Prove the identity whenever n, r , and k are nonnegative
Chapter 9, Problem 22E(choose chapter or problem)
Prove the identity \(\left(\begin{array}{l}n \\ 7\end{array}\right)\left(\begin{array}{l}7 \\ k\end{array}\right)=\left(\begin{array}{c}n \\ k\end{array}\right)\left(\begin{array}{c}n-k \\ 7-k\end{array}\right)\), whenever \(n, r\), and \(k\) are nonnegative integers with \(r \leq n\) and \(k \leq r\),
a) using a combinatorial argument.
b) using an argument based on the formula for the number of r-combinations of a set with \(n\) elements.
Equation Transcription:
r ≤ n
k ≤ r
Text Transcription:
(_r^n) (_k^r)=(_k^n)(_r-k^n-k),
r leq n
k leq r
r
n
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