what are the coefficients for the binomial expansion of (p + q)^6?\n1, 8, 28, 56, 70, 56, 28, 8, 1\n1, 6…

Explanation:

Step1: Recall binomial theorem coefficients

The coefficients of the binomial expansion ((a + b)^n) can be found using the (n) -th row of Pascal's triangle or the formula (C(n,k)=\frac{n!}{k!(n - k)!}), where (n) is the power of the binomial and (k) ranges from (0) to (n). For ((p+q)^6), (n = 6).

Step2: Calculate coefficients using formula

When (k = 0), (C(6,0)=\frac{6!}{0!(6 - 0)!}=\frac{6!}{6!}=1). When (k = 1), (C(6,1)=\frac{6!}{1!(6 - 1)!}=\frac{6!}{1!5!}=6). When (k = 2), (C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5}{2\times 1}=15). When (k = 3), (C(6,3)=\frac{6!}{3!(6 - 3)!}=\frac{6\times5\times4}{3\times2\times1}=20). When (k = 4), (C(6,4)=C(6,2) = 15) (since (C(n,k)=C(n,n - k))). When (k = 5), (C(6,5)=C(6,1)=6). When (k = 6), (C(6,6)=C(6,0)=1).

Answer:

1, 6, 15, 20, 15, 6, 1