11th Class Mathematics Permutations and Combinations Permutation & Combination

Permutation & Combination

Category : 11th Class

PERMUTATION & COMBINATION

Learning Objectives

Factorial
Permutation
Combination

Factorial

The factorial, symbolized by an exclamation mark (!), is a quantity defined for all integers greater than or equal to 0. Mathematically, the formula for the factorial is as follows.

If n is an integer greater than or equal to 1, then

n! = n (n - 1) (n - 2) (n - 3) …. (3)(2)(10).

Example:

\[1!=1,\,\,2!=2,\,\,3!=6,\,\,4!=4.3.2.1=24,\,\,5!=5\times 4\times 3\times 2\times 1=120\]

\[61=6\times 5\times 4\times 3\times 2\times 1=720,\text{ }7!=\text{ }5040\text{ }and\text{ }8!=40320\text{ }etc.\]

The special case 0! is defined to have value 0! = 1.

Permutation

The different arrangements which can be made by taking some or all of the given things or objects at a time is called Permutation.

All permutations (arrangements] made with the letters a, b, c by taking two at a time will be (ab, be, ca, ba, ac, cb).

Number of Permutations: Number of all permutations of n things, taking r at a time is:

\[^{n}{{P}_{r}}=\frac{n!}{n-r!}=n(n-1)\,(n-2)\,(n-3)\,...\,\,...\,(n-r+1)\]

Note: This is valid only when repetition is not allowed.

Permutation of n different things taken rat a time. When repetition is Allowed: \[n\times \,\,n\times n\,\,...........\,r\text{ }times=\,{{n}^{r}}\] ways
Permutation of n things taking all n things at a time = n!
Out of n objects \[{{n}_{1}}\] are alike one type, \[{{n}_{2}}\] are alike another type, \[{{n}_{3}}\] are alike third type, \[nr\] nr are alike another type such that \[({{n}_{1}}+{{n}_{2}}+{{n}_{3}}\,......\,nr)=n\]

Number of permutations of these n things are \[=\frac{n!}{{{n}_{1}}!\,{{n}_{2}}!\,\,...\,\,...\,\,.{{n}_{r}}!}\]

Combination

Each of the different selections or groups which are made by taking some or all of a number of things or objects at a time is called combination.

The number of combinations of n dissimilar things taken r at a time is denoted by \[^{n}{{C}_{r}}\,or\,C\,(n,\,r).\]

\[^{n}{{C}_{r}}\,=\frac{n!}{r!\,(n-r)!}=\frac{n\,(n-1)\,(n-2)\,.....\,(n-r+1)}{1.2.3......r}\]

Also \[^{n}{{C}_{0}}=1;\,{{}^{n}}{{C}_{n}}=1;\]

\[Note:\,\,(i){{\,}^{n}}{{C}_{r}}+{{\,}^{n}}{{C}_{r-1}}=\,{{\,}^{(n+1)}}{{C}_{r}}\]

Important Formula

In a group of n-members if each member offers a shake hand to the remaining members then the total number of handshakes \[^{n}{{C}_{2}}=\frac{n\,(n-1)}{1.2}=\frac{n\,(n-1)}{2}\]
The number of diagonals in a regular polygon of 'n' sides is \[\frac{n\,(n-3)}{2}\]
From a group of m-men and n-women, if a committee of remembers \[(r\,\le \,m+n)\] to be formed, then the number of ways it can be done is equal to \[^{(m+n)}{{C}_{r}}.\]
The number of ways a group of r-boys (men) and s-girls (women) can be made out of m boys (men) and n-girls (women) is equal to \[{{(}^{m}}{{C}_{r}}\times \,{{\,}^{n}}{{C}_{s}}).\]
From a group of m-boys and n-girls the number of different ways that a committee of remembers can be formed so that the committee will have at least one girl is \[{{C}_{r}}-{{\,}^{m}}{{C}_{r}}\]

Commonly Asked Questions

If a die is cast and then a coin is tossed, find the number of all possible outcomes.

(a) 11 (b) 12

(e) None of these

Answer: (b)

Explanation: A die can fall in 6 different ways showing six different points 1, 2, 3, 4, 5, 6,... and a coin can fall in 2 different ways showing head (H) or tail (T).

\[\therefore \] The number of all possible outcomes from a die and a coin \[=\text{ }6\times 2=12.\]

There are 6 trains running between indore and Bhopal. In how many ways can a man go from indore to Bhopla and by a different train?

(a) 25 (b) 35

(e) None of these

Answer (c)

Explanation: A man can go from Indore to Bhopal in 6 ways by any one of the fc trains available. Then he can return from Bhopal to Indore in 5 ways by the remaining 3 trains, since he cannot return by the same train by which he goes to Bhopal from Indore.

Thus, the required number of ways \[=\text{ }6\times 5=30.\]

Find the value of 9?

(a) 504 (b) 309

(e) None of these

Answer (a)

Explanation:

\[^{9}{{P}_{30}}=\frac{9!}{6!}\left( \therefore \,{{\,}^{n}}{{P}_{r}}=\frac{n!}{(n-r)!} \right)\]

\[=\frac{9\times 8\times 7\times 6!}{6!}=9\times 8\times 7=504\]

In how many different ways can the letters of the word 'stress' be arranged?

(a) 120 (b) 420

(e) None of these

Answer: (a)

Explanation: The word 'STRESS' has a total of six Setters (n = 6) out of which a group of three letters are same (a = 3)

\[\therefore \] The letters can be arranged in \[\frac{n!}{a!}=\frac{6!}{3!}=\frac{720}{6}=120\].

Find the value of \[^{8}{{C}_{3}}\]

(a) \[56\] (b) \[8!\]

(e) None of these

Answer: (a)

Explanation: \[^{8}{{C}_{3}}=\frac{8!}{3!\,.\,5!}=\frac{8\times 7\times 6}{6}=56\]

Directions: Study the given information carefully and answer the questions that follow:

A committee of five members is to be formed out of 3 trainees, 4 professors and 6 research associates. In how many different ways can this be done if-

The committee should have all 4 professors and 1 research associate or all 3 trainees and 2 professors?

(a) 13 (b) 12

(e) None of these

Answer: (b)

Explanation: Five member team with 4 professors and 1 research associate can be selected in \[^{4}{{C}_{4}}\times \,{{\,}^{6}}{{C}_{1}}=1\times 6=6\] ways. Five member team with 3 trainees and 2 professors can be selected in \[^{3}{{C}_{3}}\times \,{{\,}^{4}}{{C}_{1}}=1\times 6=6\] ways.

\[\therefore \] Total number of ways of selecting the committee = 6 + 6 = 12.

The committee should have 2 trainees and 3 research associates?

(a) 15 (b) 45

(e) None of these

Answer: (c)

Explanation: 2 trainees and 3 research associates can be selected in \[^{3}{{C}_{2}}\times \,{{\,}^{6}}{{C}_{3}}=3\times \,20=60\] ways.

11th Class Mathematics Permutations and Combinations Permutation & Combination

Other Topics

Notes - Permutation & Combination