document.write( "Question 1046042: Let $\"N+=+N%5B1%5D+%2B+N%5B2%5D\"$ , and $\"1+%3C=+k+%3C=+N%5B1%5D\"$ and $\"1+%3C=+k+%3C=+N%5B2%5D\"$ .\r
\n" ); document.write( "\n" ); document.write( "Prove that\r
\n" ); document.write( "\n" ); document.write( "+...+ $\"C%28N%5B1%5D%2C1%29%2AC%28N%5B2%5D%2Ck-1%29+%2B+C%28N%5B1%5D%2C0%29%2AC%28N%5B2%5D%2Ck%29\"$ . \n" ); document.write( "

Algebra.Com's Answer #661747 by robertb(5830) $\"\"$ $\"About$

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Use a combinatorial approach.\r
\n" ); document.write( "\n" ); document.write( "Suppose a group of N people consists of two groups, with group A having $\"N%5B1%5D\"$ people and group B having $\"N%5B2%5D\"$ people, and that $\"N=+N%5B1%5D+%2B+N%5B2%5D\"$ .\r
\n" ); document.write( "\n" ); document.write( "The number of ways of selecting 0 people from group A and selecting k people from group B is $\"%28matrix%282%2C1%2C+N%5B1%5D%2C0%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck%29%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "The number of ways of selecting 1 person from group A and selecting k-1 people from group B is $\"%28matrix%282%2C1%2C+N%5B1%5D%2C1%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck-1%29%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "The number of ways of selecting 2 people from group A and selecting k-2 people from group B is $\"%28matrix%282%2C1%2C+N%5B1%5D%2C2%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck-2%29%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "This goes on until we come to the number of ways of selecting k people from group A and selecting 0 people from group B which is $\"%28matrix%282%2C1%2C+N%5B1%5D%2Ck%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2C0%29%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "The total number of ways of selecting k people from N people, based on membership on group A or B, is then\r
\n" ); document.write( "\n" ); document.write( " $\"%28matrix%282%2C1%2C+N%5B1%5D%2C0%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck%29%29\"$ + $\"%28matrix%282%2C1%2C+N%5B1%5D%2C1%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck-1%29%29\"$ + $\"%28matrix%282%2C1%2C+N%5B1%5D%2C2%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck-2%29%29\"$ +...+ $\"%28matrix%282%2C1%2C+N%5B1%5D%2Ck%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2C0%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "But the total number of ways of selecting k people from N people, without any restriction (or regardless of membership), is $\"%28matrix%282%2C1%2C+N%2Ck%29%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "Therefore,\r
\n" ); document.write( "\n" ); document.write( " $\"%28matrix%282%2C1%2C+N%2Ck%29%29\"$ = $\"%28matrix%282%2C1%2C+N%5B1%5D%2C0%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck%29%29\"$ + $\"%28matrix%282%2C1%2C+N%5B1%5D%2C1%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck-1%29%29\"$ + $\"%28matrix%282%2C1%2C+N%5B1%5D%2C2%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2Ck-2%29%29\"$ +...+ $\"%28matrix%282%2C1%2C+N%5B1%5D%2Ck%29%29%2A%28matrix%282%2C1%2C+N%5B2%5D%2C0%29%29\"$ . \n" ); document.write( "

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