Find the exact value of (90 choose 0)-(90 choose 1)+(90 choose 2)-(90 choose 3)+...-(90 choose 87)+(90 choose 88)

Question

I want to know if I have to determine the missing terms and then compute the whole thing?

Answer 1

We can "complete the pattern" by adding the quantity $-\binom{90}{89} + \binom{90}{90}$, making sure we take them back out later. This means we can represent the summation as follows:

$$S = \left(\sum_{k=0}^{90} \binom{90}{k}(-1)^k\right) + \binom{90}{89} - \binom{90}{90}$$

And by the binomial theorem, which states that $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^{k}$, the sum we wish to compute has $n=90, x=1, y=-1$, which implies:

$$S = (1-1)^{90} + 90 - 1 = 89$$

For more detail see the link from Wikipedia.