Question

Give a combinatorial proof of one of the following: (a) For $n ge k ge 3$, $k(k-1)(k-2) inom{n}{k} = n(n-1)(n-2) inom{n-3}{k-3}$. (b) For $n ge 1$, $sum_{k=0}^{n} k inom{n}{k} = n2^{n-1}$. (c) For $n ge k ge 1$, $S(n, k) = S(n-1, k-1) + k cdot S(n-1, k)$.

          Give a combinatorial proof of one of the following:
(a) For $n ge k ge 3$,
$k(k-1)(k-2) inom{n}{k} = n(n-1)(n-2) inom{n-3}{k-3}$.
(b) For $n ge 1$,
$sum_{k=0}^{n} k inom{n}{k} = n2^{n-1}$.
(c) For $n ge k ge 1$,
$S(n, k) = S(n-1, k-1) + k cdot S(n-1, k)$.

Give a combinatorial proof of one of the following:
(a) For n ge k ge 3,
k(k-1)(k-2) inomnk = n(n-1)(n-2) inomn-3k-3.
(b) For n ge 1,
sumk=0^n k inomnk = n2^n-1.
(c) For n ge k ge 1,
S(n, k) = S(n-1, k-1) + k cdot S(n-1, k).

Added by Sara H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Hoan Nguyen

05/19/2023

Step 1

(a) For n k 3, k(k 1)(k 2) ( =n(n 1)(n = 2)( Show more…

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Give a combinatorial proof of one of the following: a) For n ≥ k ≥ 3, k(k-1)(k-2)(n choose k) = n(n-1)(n-2)(n-3 choose k-3) b) For n ≥ 1, ̳(k=0 to n) k(n choose k) = n2^(n-1) c) For n ≥ k ≥ 1, S(n,k) = S(n-1,k-1) + kS(n-1,k)