Pigeon Hole explanation












2












$begingroup$


I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.



Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).



Answer of this question being $97$ baskets.






pigeonhole-principle







share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.



    Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).



    Answer of this question being $97$ baskets.






    pigeonhole-principle







    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.



      Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).



      Answer of this question being $97$ baskets.






      pigeonhole-principle







      share|cite|improve this question











      $endgroup$




      I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.



      Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).



      Answer of this question being $97$ baskets.






      pigeonhole-principle




      pigeonhole-principle






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 18 hours ago









      Peter

      49.2k1240138




      49.2k1240138










      asked 18 hours ago









      LaykenLayken

      324




      324






















          4 Answers
          4






          active

          oldest

          votes


















          3












          $begingroup$

          The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



          Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



          The problem here :



          There are $24$ possibilities for the number of apples in a basket.



          Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



          But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.






          share|cite|improve this answer









          $endgroup$





















            3












            $begingroup$

            Why $96$ is not enought:



            For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



            On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.






            share|cite|improve this answer









            $endgroup$





















              2












              $begingroup$

              Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.






              share|cite|improve this answer









              $endgroup$





















                2












                $begingroup$

                The number of apples in a basket is between 1 and 24. That's 24 different values.



                Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                This means that in our collection of baskets each of these numbers occurs at most four times.



                But then we can have at most 24 times 4 baskets.



                Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.






                share|cite|improve this answer











                $endgroup$













                • $begingroup$
                  Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                  $endgroup$
                  – L. F.
                  17 hours ago












                Your Answer








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                4 Answers
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                active

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                4 Answers
                4






                active

                oldest

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                active

                oldest

                votes






                active

                oldest

                votes









                3












                $begingroup$

                The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



                Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



                The problem here :



                There are $24$ possibilities for the number of apples in a basket.



                Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



                But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.






                share|cite|improve this answer









                $endgroup$


















                  3












                  $begingroup$

                  The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



                  Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



                  The problem here :



                  There are $24$ possibilities for the number of apples in a basket.



                  Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



                  But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.






                  share|cite|improve this answer









                  $endgroup$
















                    3












                    3








                    3





                    $begingroup$

                    The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



                    Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



                    The problem here :



                    There are $24$ possibilities for the number of apples in a basket.



                    Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



                    But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.






                    share|cite|improve this answer









                    $endgroup$



                    The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



                    Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



                    The problem here :



                    There are $24$ possibilities for the number of apples in a basket.



                    Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



                    But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 18 hours ago









                    PeterPeter

                    49.2k1240138




                    49.2k1240138























                        3












                        $begingroup$

                        Why $96$ is not enought:



                        For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



                        On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.






                        share|cite|improve this answer









                        $endgroup$


















                          3












                          $begingroup$

                          Why $96$ is not enought:



                          For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



                          On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.






                          share|cite|improve this answer









                          $endgroup$
















                            3












                            3








                            3





                            $begingroup$

                            Why $96$ is not enought:



                            For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



                            On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.






                            share|cite|improve this answer









                            $endgroup$



                            Why $96$ is not enought:



                            For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



                            On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 18 hours ago









                            Nathanael SkrepekNathanael Skrepek

                            1,7921615




                            1,7921615























                                2












                                $begingroup$

                                Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.






                                share|cite|improve this answer









                                $endgroup$


















                                  2












                                  $begingroup$

                                  Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.






                                  share|cite|improve this answer









                                  $endgroup$
















                                    2












                                    2








                                    2





                                    $begingroup$

                                    Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.






                                    share|cite|improve this answer









                                    $endgroup$



                                    Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 18 hours ago









                                    tonychow0929tonychow0929

                                    42137




                                    42137























                                        2












                                        $begingroup$

                                        The number of apples in a basket is between 1 and 24. That's 24 different values.



                                        Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                                        This means that in our collection of baskets each of these numbers occurs at most four times.



                                        But then we can have at most 24 times 4 baskets.



                                        Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.






                                        share|cite|improve this answer











                                        $endgroup$













                                        • $begingroup$
                                          Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                          $endgroup$
                                          – L. F.
                                          17 hours ago
















                                        2












                                        $begingroup$

                                        The number of apples in a basket is between 1 and 24. That's 24 different values.



                                        Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                                        This means that in our collection of baskets each of these numbers occurs at most four times.



                                        But then we can have at most 24 times 4 baskets.



                                        Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.






                                        share|cite|improve this answer











                                        $endgroup$













                                        • $begingroup$
                                          Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                          $endgroup$
                                          – L. F.
                                          17 hours ago














                                        2












                                        2








                                        2





                                        $begingroup$

                                        The number of apples in a basket is between 1 and 24. That's 24 different values.



                                        Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                                        This means that in our collection of baskets each of these numbers occurs at most four times.



                                        But then we can have at most 24 times 4 baskets.



                                        Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.






                                        share|cite|improve this answer











                                        $endgroup$



                                        The number of apples in a basket is between 1 and 24. That's 24 different values.



                                        Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                                        This means that in our collection of baskets each of these numbers occurs at most four times.



                                        But then we can have at most 24 times 4 baskets.



                                        Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.







                                        share|cite|improve this answer














                                        share|cite|improve this answer



                                        share|cite|improve this answer








                                        edited 5 hours ago

























                                        answered 18 hours ago









                                        jflippjflipp

                                        3,7711711




                                        3,7711711












                                        • $begingroup$
                                          Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                          $endgroup$
                                          – L. F.
                                          17 hours ago


















                                        • $begingroup$
                                          Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                          $endgroup$
                                          – L. F.
                                          17 hours ago
















                                        $begingroup$
                                        Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                        $endgroup$
                                        – L. F.
                                        17 hours ago




                                        $begingroup$
                                        Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                        $endgroup$
                                        – L. F.
                                        17 hours ago


















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                                        Tu-95轟炸機

                                        Image
                                        body.skin-minerva .mw-parser-output table.infobox caption{text-align:center} 图-95 Ту–95 北约代号：熊（Bear） 一架在安格斯空军基地的图-95MS 概觀 類型 战略轰炸机 代號 北约代号： Bear （熊） 乘員 7名 駕駛員2名，機尾炮手1名，其他人員4名 首飛 1952年11月12日 服役 1956年 設計 圖波列夫設計局 產量 超过500架 現況 现役 主要用戶   蘇聯   俄羅斯 衍生機型 图-114、图-119、Tu-95 技术数据 長度 49.50米（162呎5吋） 翼展 51.10米（167呎8吋） 高度 12.12米（39呎9吋） 翼面積 310平方米（3,330平方呎） 空重 90,000公斤（198,000磅） 最大起飛重量 188,000公斤（414,500磅） 發動機 4具库兹涅佐夫NK-12MV型渦輪螺旋槳發動機 功率 4×11,000千瓦（14,800馬力） 性能數據 最大速度 925公里／時（500節，575哩／時） 爬升率 10米／秒（2,000呎／分） 最大升限 12,000米（39,000呎） 最大航程 15,000公里（8,100海浬，9,400哩） 翼負荷 606公斤／平方米（124磅／平方呎） 推重比 235 W／公斤（0.143馬力／磅） 武器装备 機炮 1具/2具AM-23型23公釐雷達控制機砲（機尾） 飛彈 空对地导弹： Kh-20、Kh-22 Kh-26、Kh-55 炸彈 60枚FAB-250炸彈，或 30枚FAB-500炸彈。 其他 载弹量（包括飞弹）：最多15,000公斤（33,000磅） Tu-95熊式MR型機 图-95 （俄语： Ту–95 ， 英语： Tu-95 ），北約代號： Bear （ 熊 ），是蘇聯圖波列夫設計局所研製，是全世界唯一服役的大型四渦輪螺旋槳發動機...
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