# 67. Permutations and Combinations Solutions Part 7¶

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.