Permutation and Combinations MATHS - Permutation and Combinations

A five digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetitions. The total number of ways this can be done is.

1. 216
2. 600
3. 240
4. 3125

[Hint: 5 digit numbers can be formed using digits 0, 1, 2, 4, 5 or by using digits 1, 2, 3, 4, 5 since sum of digits in these cases is divisible by 3.]

Given 5 different green dyes, four different blue dyes and three different red dyes, the number of combinations of dyes which can be chosen taking at least one green and one blue dye is.

1. 3600
2. 3720
3. 3800
4. 3600

[Hint: Possible numbers of choosing or not choosing 5 green dyes, 4 blue dyes and 3 red dyes are 2⁵ , 2⁴ and 2³ , respectively.]

Every body in a room shakes hands with everybody else. The total number of hand shakes is 66. The total number of persons in the room is.

1. 11
2. 12
3. 13
4. 14

If ⁿC₁₂ = ⁿC₈ , then n is equal to.

1. 20
2. 12
3. 6
4. 30

The total number of 9 digit numbers which have all different digits is.

1. 10!
2. 9 !
3. 9 × 9!
4. 10×10!

In the permutations of n things, r taken together, the number of permutations in which m particular things occur together is ^n–mP_r–m × ^rP_m .

There will be only 24 selections containing at least one red ball out of a bag containing 4 red and 5 black balls. It is being given that the balls of the same colour are identical.

Three letters can be posted in five letterboxes in 35 ways.

There are 12 points in a plane of which 5 points are collinear, then the number of lines obtained by joining these points in pairs is ¹²C₂ – ⁵C₂ .

In a football championship, 153 matches were played . Every two teams played one match with each other. The number of teams, participating in the championship is ______.

The total number of ways in which six ‘+’ and four ‘–’ signs can be arranged in a line such that no two signs ‘–’ occur together is ______.

A committee of 6 is to be chosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee.
[Hint: At least 3 men and 2 women: The number of ways = ¹⁰C₃ × ⁷C₃ + ¹⁰C₄ × ⁷C₂ . For 2 particular women to be always there: the number of ways = ¹⁰C4 + ¹⁰C₃ × ⁵C₁ . The total number of committees when two particular women are never together = Total – together.]

Three balls are drawn from a bag containing 5 red, 4 white and 3 black balls. The number of ways in which this can be done if at least 2 are red is ______.

A box contains two white, three black and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw.
[Hint: Required number of ways = ³C₁ × ⁶C₂ + ³C₂ × ⁶C₂ + ³C₃ .]

Eight chairs are numbered 1 to 8. Two women and 3 men wish to occupy one chair each. First the women choose the chairs from amongst the chairs 1 to 4 and then men select from the remaining chairs. Find the total number of possible arrangements.
[Hint: 2 women occupy the chair, from 1 to 4 in ⁴P₂ ways and 3 men occupy the remaining chairs in ⁶P₃ ways.]

How many committee of five persons with a chairperson can be selected from 12 persons.
[Hint: Chairman can be selected in 12 ways and remaining in 11C₄ .]

If the letters of the word RACHIT are arranged in all possible ways as listed in dictionary. Then what is the rank of the word RACHIT?
[Hint: In each case number of words beginning with A, C, H, I is 5!]

In an examination, a student has to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.

Match each item given under the column C₁ to its correct answer given under the column C₂.
How many words (with or without dictionary meaning) can be made from the letters of the word MONDAY, assuming that no letter is repeated, if

	C1		C2
(a)	4 letters are used at a time.	(i)	720
(b)	All letters are used at a time.	(ii)	240
(c)	All letters are used but the first is a vowel.	(iii)	360

A sports team of 11 students is to be constituted, choosing at least 5 from Class XI and atleast 5 from Class XII. If there are 20 students in each of these classes, in how many ways can the team be constituted?

A candidate is required to answer 7 questions out of 12 questions, which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. Find the number of different ways of doing questions.

A convex polygon has 44 diagonals. Find the number of its sides.
[Hint: Polygon of n sides has (ⁿC₂ – n) number of diagonals.]

Match each item given under the column C₁ to its correct answer given under the column C₂.
Five boys and five girls form a line. Find the number of ways of making the seating arrangement under the following condition:

	C1		C2
(a)	Boys and girls alternate.	(i)	5! × 6!
(b)	No two girls sit together.	(ii)	10! – 5! 6!
(c)	All the girls sit together.	(iii)	(5!)2 + (5!)2
(d)	All the girls are never together.	(iv)	2! 5! 5!

Match each item given under the column C₁ to its correct answer given under the column C₂.
Using the digits 1, 2, 3, 4, 5, 6, 7, a number of 4 different digits is formed. Find

	C1		C2
(a)	how many numbers are formed.	(i)	840
(b)	how many numbers are exactly divisible by 2.	(ii)	200
(c)	how many numbers are exactly divisible by 25.	(iii)	360
(d)	how many of these are exactly divisble by 4.	(iv)	40

Match each item given under the column C₁ to its correct answer given under the column C₂.
There are 10 professors and 20 lecturers out of whom a committee of 2 professors and 3 lecturer is to be formed. Find:

	C1		C2
(a)	In how many ways committee can be formed.	(i)	10C2 × 19C3
(b)	In how many ways a particular professor is included.	(ii)	10C2 × 19C2
(c)	In how many ways a particular lecturer is included.	(iii)	9C1 × 20C3
(d)	In how many ways a particular lecturer is excluded.	(iv)	10C2 × 20C3

Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.