JEE Advanced 2025 Paper 2 · Mathematics · Binomial Theorem · medium

The coefficient of x^6 in the expansion of (1+x)^5(1+x+x^2^4 is ______.

Numerical answer

The coefficient of

x^6

in the expansion of

(1+x)^5(1+x+x^2)^4

is ______.

Show answer and solution

Answer: 494

Expand each factor.

(1+x)^5

has coefficients

1,5,10,10,5,1

(1+x+x^2)^4

has coefficients

1,4,10,16,19,16,10,4,1

(trinomial coefficients). The coefficient of

x^6

in the product is

\sum_{j=0}^{5}c_j\,d_{6-j}

where

c_j

are binomial and

d_i

trinomial coefficients:

1\cdot10+5\cdot16+10\cdot19+10\cdot16+5\cdot10+1\cdot4=10+80+190+160+50+4=494

. A computer-algebra expansion of the product gives the same coefficient,

494

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