Question

How to solve examples c) d) ?
Factorial sums

Answer 1

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.