Question

9. (99)

Answer 1

The given expression ( 99 ) 5 .

The formula for binomial expansion is ,

. ( a+b ) n = C n 0 a n + C n 1 a n−1 b+ C n 2 a n−2 b 2 +..........+ C n n−1 a. b n−1 + C n n b n (1)

To apply the binomial theorem, power of expression is easier to calculate so we can write sum or difference of two number which is equal to 101 .

99=100−1 (2)

On comparing the equations (1) and (2), values of a=100, b=−1 and n=5 .

( 99 ) 5 = ( 100−1 ) 5 = C 5 0 ( 100 ) 5 + C 5 1 ( 100 ) 4 ( −1 ) 1 + C 5 2 ( 100 ) 3 ( 1 ) 2 + C 5 3 ( 100 ) 2 ( −1 ) 3 + C 5 4 ( 100 ) 1 ( 1 ) 4 + C 5 5 ( −1 ) 5 = ( 100 ) 5 −5 ( 100 ) 4 +10 ( 100 ) 3 10 ( 100 ) 2 +5( 100 )−1 =10000000000−500000000+1000000−100000+500−1 =10010000500−500100001 =9509900499

Thus ( 99 ) 5 is equal to 9509900499 .