The principle of multiplication states that “If we can perform a particular operation in ‘n’ ways and the second operation in ‘m’ ways, the two operations can be performed in m x n ways in succession”. This is applicable to a finite number of operations. A 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 | 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 the 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, and 4 oranges among 4 persons if each can receive any number of fruits and the same type of fruits is identical.
Solution:
Here, p = 4
Using the formula from the 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, and 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 the Beggar’s method, n+p-1Cp-1
= 7+3-1C3-1
= 9C2
Beggar’s Method – Video Lesson
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Frequently Asked Questions
When do we use the Beggar’s method?
The Beggar’s Method is used for the 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 nCr = n!/r!(n-r)!.
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