Permutations and Combinations Formula

Factorial

For any natural number n, we define factorial as

n! = n(n-1)(n-2)…3×2×1

Important Results Related to Factorial

1. 0! = 1! = 1
2. Factorials of negative integers and fractions are not defined.
3. n! = n(n-1)! = n(n-1)(n-2)!

Permutations

The number of ways of arranging n distinct objects taking some or all of a number of things is called a permutation.

P(n, r) = ⁿP_r = n!\over (n-r)!

Properties of permutation

1. ⁿP_n = n(n-1)(n-2) …. 3×2×1
2. ⁿP_o = n!\over n!
3. ⁿP_o = n
4. ⁿP_{n – 1} = n!
5. ^{n – 1} P_r + r. ^{n – 1} P_{r – 1} = ⁿP_r
6. {{^nP_r}\over{^nP_{r-1}}}=n-r+1

Important Results on Permutation

1. The number of permutations of n different things taken r at a time, when each thing may be repeated any number of times is n^r.
2. The number of permutations of n things taken all at a time, in which p are alike of one kind, q are alike of the second kind and r are alike of the third kind and rest are different is
   n!\over p!q!r!

Restricted Permutations

(i) Number of permutations of n different things taken r at a time,

1. When a particular thing is to be included in each arrangement is r.^n-1P_r-1.
2. When a particular thing is always excluded is ^n-1P_r.

(ii) Number of permutations of n different objects taken r at a time in which m particular object are always

1. excluded = ^n-mP_r
2. included = ^n-mP_r-m × r!

(iii) Number of permutations of n different things taken all at a time, when m specified things always come together is m!(n – m + 1)!

Circular Permutations

A circular permutation is an arrangement of objects in a circle, where the order of the objects is important but the starting position is not.

1. The number of circular permutations of n different things taken all at a time is (n-1)!. If clockwise and anticlockwise orders are taken as different.
2. The number of circular permutations of n different things taken all at a time when clockwise and anticlockwise orders are not taken as different = {1\over2} (n-1)!

Combination

A combination is a selection of items from a set where the order of the selection does not matter.

C (n, r) or ⁿC_r = n!\over r!(n - r)! 0≤ r ≤ n

Properties of combination

1. ⁿC_o = ⁿC_n = 1
2. ⁿC₁ = n
3. ⁿC_r = ⁿC_n-r
4. If ⁿC_a = ⁿC_b, then either a = b or a + b = n
5. ⁿC_r = ^nP_r\over r!
6. ⁿC_r + ⁿC_{r – 1} = ⁿ⁺¹C_r
7. { {^n}C_r \over{^{n-1}}C_{r-1}}={n\over r}
8. { {^n}C_r \over{^{n}}C_{r-1}}={n-r+1\over r}
9. ⁿC₀ + ⁿC₁ + ⁿC₂ + ⁿC₃ + … + ⁿC_n = 2ⁿ
10. ⁿC₀ + ⁿC₂ + ⁿC₄ + … = ⁿC₁ + ⁿC₃ + ⁿC₅ + … = 2^n-1

The important result on the combination

1. The number of combinations of n different things have taken r at a time allowing repetitions is ^n+r-1C_r.
2. The total number of combinations of n different objects taken r at a time in which
   (a) m particular objects are excluded = ^n-mC_r
   (b) m particular objects are included = ^n-mC_r-m

Selection from different Items

1. The number r of ways of selecting at least one item from n distinct items is 2^n-1.
2. The number of ways of answering one or more of n questions is 2ⁿ – 1.
3. The number of ways of answering one or more of n questions when each question has an alternative = 3^n-1.

Division into groups

1. The number of ways in which (n+m) different things can be divided into two groups which m and n things respectively = (m+n)!\over m!n!, where m ≠ n
2. The number of ways of dividing 2n different elements into two groups of n objects each is (2n)!\over (n!)^2, when the distinction can be made between the groups.
3. The number of ways of dividing 2n different elements into two groups of n objects when no distinction can be made between the groups = (2n)!\over 2(n!)^2