If 1 + x + x 2 8 = a 0 + a 1 x + a 2 x 2 + … + a 16 x 16 for all values of x , then a 5 100 is equal to

Question

If 1 + x + x 2 8 = a 0 + a 1 x + a 2 x 2 + … + a 16 x 16 for all values of x , then a 5 100 is equal to,

MARKS App · Accepted Answer

Coefficient of x 5 = choosing x from 5 brackets out of 8 or choosing one x 2 and three x or choosing two x 2 and one x = C 8 5 + C 8 1 C 7 3 + C 8 2 C 6 1 = 8 × 7 × 6 3 × 2 + 8 × 7 × 6 × 5 3 × 2 + 8 × 7 × 6 2 = 56 + 56 × 5 + 56 × 3 = 56 × 9 = 504 ⇒ a 5 = 504

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