Some Important Results

1. The coefficient of xr in the expansion of (1 − x)−n = n + ^{r −1}Cr.
2. If there are k objects of one kind, l objects of another kind and so on; then the number of ways of choosing r objects out of these objects (k + l….) is the coefficient of xr in (1 + x + x² + .. + x^k) (1 + x +x ² + … + xl) ….
3. If there are k objects of one kind, l objects of another kind and so on, then the number of possible arrangements of r objects out of these objects is (l + k + ….) is
   Coefficient of xr in r! × ….

Example

How many words of four letters can be formed from the letters of the word "EXAMINATION"

Solution

There are 11 letters AAIINNEXMTO

Coefficient of x⁴ in 4! (1 + x)⁵ = 4! = 2454