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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

Beggar’s Method – Video Lesson

Beggar's Method

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-1Cp-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 nCr = n!/r!(n-r)!.

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