# 65. Permutations and Combinations Problems Part 7¶

1. If $$ab = n(a>1, b>1),$$ then show that $$(n - 1)!$$ divisible by both $$a$$ and $$b.$$

2. Show that $$(kn)!$$ is divisible by $$(n!)^k$$

3. In how many ways $$20$$ apples be distributed among $$5$$ persons if each person can get any number of things.

4. In how many ways $$r$$ flags be displayed on $$n$$ poles in a row, disregarding the limitation on the number of flags on a pole?

5. If $$x + y + z = n,$$ where $$x, y, z, n$$ are positive integers, find the number of integral solution of this equation.

6. Find the number of integral solutions of $$x + y + z = 0$$ with $$x, y, z \geq -5$$

7. In an examination, the maximum marks for each of the three papers is $$n$$; for the fourth paper it is $$2n.$$ Prove that the number of ways in which a student can get $$3n$$ marks is $$\frac{1}{6} (n + 1)(5n^2 + 10n + 6)$$

1. Find the number of positive integral solutions of the equation $$x_1 + x_2 + x_3 = 10$$

2. Find the number of non-negative integral solutions of equation $$3x + y + z = 24$$

3. Find the number of integral solutions of the equation $$x + y + z + t = 29,$$ where $$x\geq 1, y\geq 2, z\geq 3, t\geq 0$$

4. Find the number of non-negative integral solution of $$a + b + c + d = 20$$

5. Find the number of non-negative integral solutions of equation $$x_1 + x_2 + \ldots + x_k \leq n$$

6. Find the number of non-negative integral solution of $$2x + 2y + z = 10$$

7. How many sets of $$2$$ and $$3$$ (different) numbers can be formed by using numbers between $$0$$ and $$180$$ (both inclusive) so that their average is $$60.$$

8. If combinations of letters be formed by taking only $$5$$ at a time out of the letters of the word METAPHYSICS, in how many of them will the letter T occur?

9. How many selections and arrangements of $$4$$ letters can be made from the letters of the word ‘proportion’?

10. A five letter word is formed such that the letter appearing in the odd numbered positions are taken from the letters which appear without repetitions in the word MATHEMATICS. Further, the letters appearing in the even numbered positions are taken from the letter which appear with repetitions in the same word MATHEMATICS. In how many different ways can the five letter word be formed?

11. Box $$1$$ contains six block lettered $$A, B, C, D, E$$ and $$F.$$ Box $$2$$ contains four block lettered $$W, X, Y$$ and $$Z.$$ How many five letter codewords can be formed by using three blocks from Box $$1$$ and two blocks from box $$2$$?

12. A tea party is arranged for $$2m$$ people along two sides of a long table with $$m$$ chairs on each side. $$r$$ men wish to sit on one particular side and $$s$$ on the other. In how many ways can they be seated? [ $$r, s\leq m$$ ]

13. A gentleman invites a party of $$10$$ friends to a dinner and there are $$6$$ places at round table and the remaining $$4$$ at another. Prove that the number of ways he can arrange them among themselves is $$151200.$$

14. A family consists of a grandfather, $$m$$ sons and daughters and $$2n$$ grandchildren. They are to be seated in a row for dinner. The grandchildren wish to occupy the $$n$$ seats at each end and grandfather refuses to have a grandchild on either side of him. In how many ways can the family be made to sit?

15. There are $$2n$$ quests at a dinner party. If the master and mistress of the house have fixed seats opposite one another and that there are two specified guests who must not be placed next to one another, find the number of ways the company can be places.

16. There are $$4n$$ things of which $$n$$ are alike and all the rest are different. Find the number of permutations of $$4n$$ things taken $$2n$$ at a time, each permutation containing the $$n$$ like things.