Mathematics: How to solve examples c) d) ? Factorial sums

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How to solve examples c) d) ? Factorial sums

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mal5546

11.11.2021

Mathematics

Recall the binomial theorem:

$\displaystyle (a+b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k$

a. Let a = b = 1 and n = 8. Then the sum above gives the sum shown here,

$\displaystyle \sum_{k=0}^8 \binom8k 1^{n-k} 1^k = \sum_{k=0}^8 \binom8k$

so it reduces to (1 + 1)⁸ = 2⁸ = 256.

b. Let a = 1 and b = -1 and k = 8. Then by the same argument, the sum reduces to (1 - 1)⁸ = 0.

c. Notice that

$\displaystyle \sum_{j=0}^n \binom n{n-j} = \binom nn + \binom n{n-1} + \binom n{n-2} + \cdots \binom n2 + \binom n1 + \binom n0$

but this is nearly identical to the sum in part a, with a = b = 1 and arbitrary n, but the order of terms is reversed. Then it reduces to (1 + 1)ⁿ = 2ⁿ.

d. Let a = 1 and b = 2. Then the sum is (1 + 2)ⁿ = 3ⁿ.

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How to solve examples c) d) ?
Factorial sums

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