Coefficient of x 11 in the expansion of (1 + x 2 ) 4 (1 + x 3 ) 7 (1 + x 4 ) 12 is
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Coefficent of x 11 ≡ 
Coefficent of x 11 ≡ (1 – x 8 ) 4 (1 + x 4 ) 8 (1 + x 3 ) 7 (1 – x 2 ) –4
= (1 – 4x 8 ) (1 + x 4 ) 8 (7x 3 + 35x 9 ) (1 – x 2 ) –4
= (7x 3 + 35x 9 – 28x 11 ) (1 + x 4 ) 8 (1 – x 2 ) –4
Coefficent of x 8 = (7x + 35x 6 – 28x 8 ) (1 + 8x 4 + 28x 8 ) (1 – x 2 ) –4
= (7 + 35x 6 – 28x 8 + 56x 4 + 196x 8 ) (1 – x 2 ) –4
Coefficent of t 4 ≡ (7 + 56t 2 + 35t 3 + 168t 4 ) (1 – t) –4
= 7 · 7 C 3 + 56 · 5 C 3 + 35 · 4 C 3 + 168
= 245 + 700 + 168 = 1113.
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