Answer: 60.
- A 60
- B 61
- C 62
- D 63
Correct answer: A. 60
Explanation: Coeff of x<sup>5</sup> in (1+x<sup>2</sup>)<sup>5</sup>(1+x)<sup>4</sup>: from (1+x<sup>2</sup>)<sup>5</sup>, only even powers x<sup>0</sup>, x<sup>2</sup>, x<sup>4</sup> are available (max x<sup>10</sup>), and from (1+x)<sup>4</sup> only powers up to x<sup>4</sup> are available. Pairing to total x<sup>5</sup>: x<sup>2</sup> from the first with x<sup>3</sup> from the second gives C(5,1)*C(4,3) = 5*4=20; x<sup>4</sup> from the first with x<sup>1</sup> from the second gives C(5,2)*C(4,1) = 10*4=40. Total = 20+40 = 60.
Row n of Pascal's triangle gives the coefficients nC0, nC1, ..., nCn for the expansion of (a+b)ⁿ; each number is the sum of the two numbers diagonally above it, a direct visual proof of the identity nCr = (n-1)C(r-1) + (n-1)Cr.
Concept context
Expansion of (a+b) n , general term, middle term, binomial coefficients, and greatest term