# Math Help - Different ways of Counting ....

1. ## Different ways of Counting ....

(a) Let n be a nonnegative integer. Use the identity

$(1 + x)^n(1 + x)^n = (1 + x)$^ $(2n)$

to show that $\sum_{n=0}^{n}\binom{n}{k}^2 =\binom{2n}{n}$ (1)

(b) Prove (1) again by counting, in two different ways, the number of ways
of choosing n people from a set of n girls and n boys.

2. Originally Posted by shmounal
(a) Let n be a nonnegative integer. Use the identity

$(1 + x)^n(1 + x)^n = (1 + x)$^ $(2n)$

to show that $\sum_{n=0}^{n}\binom{n}{k}^2 =\binom{2n}{n}$ (1)

Compare the coefficient of $x^2$ in both expressions above.

Tonio

(b) Prove (1) again by counting, in two different ways, the number of ways
of choosing n people from a set of n girls and n boys.
.