(a) \(\frac{\binom{n}{k}}{\binom{n}{k - 1}} = \frac{\frac{n!}{k! (n - k)!}}{\frac{n!}{(k - 1)! (n - k + 1)!}} = \frac{n!(k - 1)! (n - k + 1)!}{k! (n - k)! n!} = \frac{(k - 1)(n - k + 1)}{k}\)
(b)
Let the expansion of (1 + x)^n be:
(1 + x)^n = C(n, 0) + C(n, 1)x + C(n, 2)x^2 + ... + C(n, n)x^n
We know that:
a:b:c = 1:8:40
Therefore:
C(n, 0) : C(n, 1) : C(n, 2) = 1 : 8 : 40
Using the formula for the binomial coefficients, we have:
C(n, 0) = 1 C(n, 1) = n
Therefore:
C(n, 2) = 28C(n, 1) - 27C(n, 0)
Using the values we have:
C(n, 2) = 28n - 27
We know that:
C(n, 0) + C(n, 1) + C(n, 2) = 1 + 8 + 40 = 49
Therefore:
1 + n + C(n, 2) = 49
Solving this equation, we get n = 9.