Aptitude ➤ Permutations And Combinations ➤ Set 1

Let n be a positive integer. Then, factorial n, denoted n! is defined as:

n! = n(n - 1)(n - 2) ... 3.2.1.

i. We define 0! = 1.

ii. 4! = (4 x 3 x 2 x 1) = 24.

iii. 5! = (5 x 4 x 3 x 2 x 1) = 120.

The different arrangements of a given number of things by taking some or all at a time, are called permutations.

i. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).

ii. All permutations made with the letters a, b, c taking all at a time are:

( abc, acb, bac, bca, cab, cba)

Number of all permutations of n things, taken r at a time, is given by:

^{n}P_{r} = n(n - 1)(n - 2) ... (n - r + 1) = |
n! |

(n-r)! |

i.

ii.

iii. Cor. number of all permutations of n things, taken all at a time = n!.

If there are n subjects of which p

Then, number of permutations of these n objects is = |
n! |

(p_{1}!).(p_{2})!.....(p_{r}!) |

Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.

i. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.

Note: AB and BA represent the same selection.

ii. All the combinations formed by a, b, c taking ab, bc, ca.

iii. The only combination that can be formed of three letters a, b, c taken all at a time is abc.

iv. Various groups of 2 out of four persons A, B, C, D are:

AB, AC, AD, BC, BD, CD.

v. Note that ab ba are two different permutations but they represent the same combination.

The number of all combinations of n things, taken r at a time is:

^{n}C_{r} = |
n! | = | n(n - 1)(n - 2) ... to r factors | . |

(r!)(n - r)! | r! |

i.

ii.

i. ^{11}C_{4} = |
(11 x 10 x 9 x 8) | = 330. |

(4 x 3 x 2 x 1) |

ii. ^{16}C_{13} = ^{16}C_{(16-13)} = ^{16}C_{3} = |
16 x 15 x 14 | = | 16 x 15 x 14 | = 560. |

3! | 3 x 2 x 1 |

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