Prove that the co-efficient of x 15 in (1 + x +x 3 + x 4 ) n is 
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Text Solution
Verified by Experts1
(1 + x + x 3 + x 4 ) n = [(1 + x)(1 +x 3 )] n = (1 + x) n (1 + x 3 ) n
power of x in (1 + x 3 ) n expansion is multiple of 3
so possible cases to get x 15 are :-
(9n) (1 + x) n 9n(1 + x 3 ) n
⇓ ⇓
n C 15 x 15 n C 0 (x 3 ) 0 = n C 15 . n C 0 x 15
n C 12 x 12 n C 1 (x 3 ) 1 = n C 12 . n C 1 x 15
n C 12 x 9 n C 2 (x 3 ) 2 = n C 9 . n C 2 x 15
n C 6 x 6 n C 3 (x 3 ) 3 = n C 6 . n C 3 x 15
n C 3 x 3 n C 4 (x 3 ) 4 = n C 3 . n C 4 x 15
n C 0 x 0 n C 5 (x 3 ) 5 = n C 0 . n C 5 x 15
∴ coefficient of x 15 is
= n C 15 . n C 0 + n C 12 . n C 1 + n C 9 . n C 2 + n C 6 . n C 3 + n C 3 . n C 4 + n C 0 . n C 5
= 
hence prooved
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