# Beggar’s Method

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:

nPr = n! / [n – r]! (without repetition)

nPr = 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 1st 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 = 45 = 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:

nCr = [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-1Cp-1

Number of things that can be given

The number of things actually given

P1

0, 1, 2, 3 ……. n

r1

P2

0, 1, 2, 3 ……. n

r2

Pp

0, 1, 2, 3 ……. n

rp

r1 + r2 + …… + rp = n

The coefficient of xn in (1 + x + …… + xn)p

= [(1 – xn+1) / (1 – x)]p

= (1 – xn+1)p (1 – x)p

The coefficient of xn in (1 – x)-p

= p+n-1Cn

= n+p-1Cp-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 R1, R2 and R3 be the rings.

Number of ways = 43

= 64

(ii) Rings are identical

Here n = 3, p = 4

Using the formula from Beggar’s method, n+p-1Cp-1 = 3+4-1C4-1

= 6C3

= 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-1Cp-1

= [10+4-1C4-1] (apples) [5+4-1C4-1] (mangoes) [4+4-1C4-1] (oranges)

= 13C3 8C3 7C3

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 S1, S2, S3 such that each receives not less than 3 toys.

S1 + S2 + S3 = 16 —- (1)

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

So, equation (1) now becomes S1’ + S2’ + S3’ = 16 – 9 = 7

Using the formula from Beggar’s method, n+p-1Cp-1

= 7+3-1C3-1

= 9C2