The principle of multiplication states that “if we can perform a particular operation in ‘n’ ways and the second operation is ‘m’ ways, then the two operations can be performed in m x n ways in succession. This is applicable to a finite number of operations. Factorial can be defined as a function that multiplies a single number with each and every number preceding it. For example 5! = 5 * 4 * 3 * 2 * 1 = 120. The concept of permutations is the arrangement of objects in a particular order. The formula for permutation is given by:

^{n}P_{r} = n! / [n – r]! (without repetition)

^{n}P_{r} = n! / [p! q! r!]

**Example:** Find the number of ways in which 5 prizes can be distributed among 4 boys where every boy can take one or more prizes.

**Solution:**

The 1^{st} prize can be distributed to any of the 4 boys, hence it is done in 4 ways. In the same way, the second, third, fourth and fifth prizes can be given in 4 ways. Total number of ways = 4 * 4 * 4 * 4 * 4 = 4^{5} = 1024 ways

Combination refers to the selection of objects without repetition where the order doesn’t matter. The formula for the combination of n things being chosen out of r is given by:

^{n}C_{r} = [n!] / [n – r]! r!

**Beggar’s Method**

It is based on the distribution of like objects. It states that “the number of ways of distributing ‘n’ identical things among ‘p’ persons without any restriction (none, 1, 2 or all or any of the number of things can be given to one person)” = ^{n+p-1}C_{p-1}

Number of things that can be given | The number of things actually given | |
---|---|---|

P_{1} |
0, 1, 2, 3 ……. n | r_{1} |

P_{2} |
0, 1, 2, 3 ……. n | r_{2} |

P_{p} |
0, 1, 2, 3 ……. n | r_{p} |

r_{1} + r_{2} + …… + r_{p }= n

The coefficient of x^{n} in (1 + x + …… + x^{n})^{p}

= [(1 – x^{n+1}) / (1 – x)]^{p}

= (1 – x^{n+1})^{p} (1 – x)^{p}

The coefficient of x^{n} in (1 – x)^{-p}

= ^{p+n-1}C_{n}

= ^{n+p-1}C_{p-1}

**Illustration 1: **In how many ways can 3 rings be worn on 4 fingers if any number of rings can be worn on any finger?

(i) Rings are distinct

(ii) Rings are identical

**Solution:**

(i) Rings are distinct

Let R_{1}, R_{2} and R_{3 }be the rings.

Number of ways = 4^{3}

= 64

(ii) Rings are identical

Here n = 3, p = 4

Using the formula from Beggar’s method, ^{n+p-1}C_{p-1} = ^{3+4-1}C_{4-1}

= ^{6}C_{3}

= 20

**Illustration 2: **Find the number of ways of distributing 10 apples, 5 mangoes, 4 oranges among 4 persons if each can receive any number of fruits and the same type of fruits are identical.

**Solution:**

Here p = 4

Using the formula from Beggar’s method, ^{n+p-1}C_{p-1}

= [^{10+4-1}C_{4-1}] (apples) [^{5+4-1}C_{4-1}] (mangoes) [^{4+4-1}C_{4-1}] (oranges)

= ^{13}C_{3} ^{8}C_{3} ^{7}C_{3}

**Illustration 3: **Find the number of ways in which 16 identical toys are distributed among 3 students such that each receives not less than 3 toys.

**Solution: **

Let the students be S_{1}, S_{2}, S_{3} such that each receives not less than 3 toys.

S_{1} + S_{2} + S_{3} = 16 —- (1)

Distribute 3 toys to each of the students in the beginning.

So, equation (1) now becomes S_{1}’ + S_{2}’ + S_{3}’ = 16 – 9 = 7

Using the formula from Beggar’s method, ^{n+p-1}C_{p-1}

= ^{7+3-1}C_{3-1}

= ^{9}C_{2}

#### Beggar’s Method – Video Lesson

## Frequently Asked Questions

### When do we use the Beggar’s method?

Beggar’s Method is used for distribution of like objects.

### What is the number of ways of distributing n identical things among p persons without any restriction?

The number of ways of distributing ‘n’ identical things among ‘p’ persons without any restriction = ^{n+p-1}C_{p-1}.

### What do you mean by combination?

Combination denotes the selection of objects without repetition where the order does not matter.

### Give the Combination formula.

The Combination formula is given by ^{n}C_{r} = n!/r!(n-r)!.