Particular cases of Binomial Theorem

(i) Replacing b by ( − b ) , in the binomial expansion of ( a + b ) n , n ∈ N , we get ( a − b ) n n C a n b − n C a n − b + n C a n − b −· · · +( − ) r n C r a n − r b r + · · · + ( − ) n n C n a b n . Observe that the sign ‘ + ’ and ‘ − ’ appear alternately in the binomial expansion of ( a − b ) n . (ii) Replacing a by and b by x , in the binomial expansion of ( a + b ) n , we get ( + x ) n = n C + n C x + n C x + · · · + n C r x r + · · · + n C n x n . In particular, when x = , n C + n C + n C + · · · + n C n = n .

If X is a set containing n elements, then we know that n C r is the number of subsets of X having exactly r elements. So by adding n C r for r = , , , . . .

, n we get the number of subsets of X . So by using the above identity we see that a set of n elements has n subsets. (iii) ( − x ) n = n C − n C x + n C x −· · · + ( − ) r n C r x r + · · · + ( − ) n x n . In particular, when x = , n C + n C + n C + · · · = n C + n C + n C + · · · = n − Example .

Find the expansion of ( x + ) . By taking a = x , b = and n = in the binomial expansion of ( a + b ) n we get ( x + ) ( x ) + ( x ) + ( x ) + ( x ) + ( x ) + x + x + x + x + x + . Example . Evaluate .

By taking a = , b = and n = in the binomial expansion of ( a − b ) n we get ( − ) C − C + C − C + C 100000000 − 8000000 + 240000 − + 92236816 . . Particular cases of Binomial Theorem Example . Find the middle term in the expansion of ( x + y ) .

Here n = ; which is even. Thus the middle term in the expansion of ( x + y ) is the term containing x y , that is the term C x y which is equal to x y . Example . Find the middle terms in the expansion of ( x + y ) .

As n = which is odd, the terms containing x y and x y are the two middle terms. They are C x y and C x y which are equal to x y and x y . Example . Find the coefficient of x in the expansion of ( + x ) .

Let us take a = and b = x in the binomial expansion of ( a + b ) . Then, x will appear in the term containing ( x ) and nowhere else. So the term containing x is C a b = C a b = × × × × × × ( x ) = × × x .