67. Permutations and Combinations Solutions Part 7ΒΆ

  1. No. of ways of dividing \(n\) different things \(a\) groups each containing \(b\) things \(= \frac{n!}{(b!)^aa!}\) which is an integer. Thus, we can see that \((n - 1)!\) is clearly divisible \(b\).

    Similarly, we can prove that \((n - 1)!\) is divisible by \(a\)

  2. No. of ways of dividing \(kn\) different things in \(k\) groups each containing \(n\) things is \(\frac{kn!}{k!(n!)^k}\) which is an integer and thus our requirement is satisfied.

  3. Let \(r = 5, n= 20,\) then required number \(= {}^{n + r - 1}C_r\)

    \(= {}^{24}C_{20}\)

  4. \({}^{n + r - 1}C_r\)

  5. Given \(x\geq 1, y\geq 1, z\geq 1\)

    \(x + y + z = n\)

    Thus, no. of solution \(= {}^{n + 3 - 1}C_n = {}^{n + 2}C_n = {}^{n + 2}C_2\)

  6. \(x + y + z = 0 \forall x, y, z\geq -5\) can be rewritten as \(a + b + c = 15 \forall a, b, c\geq 0\)

    Thus, answer is \({}^{15 + 3 - 1}C_2 = 136\)

  7. Required number \(=\) coeff. of \(x^{3n}\) in \((1 + x + \ldots + x^n)^3(1 + x + \ldots + x^{3n})\)

Rest of the problems are left as exercises.