Alternative Proof of Binomial Theorem for Positive Integral Index (Combinatorial Method).
We have, (a + b)n = (a + b) (a + b) ....... n times.
The terms on the RHS are obtained by taking one letter from each factor and multiplying them together.
Choosing 'a' from all the factors, we get the term an.
Choosing 'a' from (n-1) factors and 'b' from the remaining factor, we get an-1b. The factor for 'b' can be chosen in nC1 ways. Thus, there are nC1 terms each equal to an-1b. Choosing 'a' from (n-2) factors and 'b' from the remaining two factors, we get an-2b2. The two factors for 'b' can be chosen in nC2 ways. Thus, there are nC2 terms each equal to an-2b2.
Proceeding in this way, we have nCk terms each equal to an-kbk.
Finally choosing 'b' from each factor, we get the term bn.
