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How many committees can be made from 15 people that have at least 2 people in each committee?

  • Last Updated : 20 Nov, 2021

In mathematics, permutation is known as the process of arranging a set in which all the members of a set are arranged into some series or order. The process of permuting is known as the rearranging of its components if the set is already arranged. Permutations take place, in more or less important ways, in almost every area of mathematics. They frequently appear when different commands on certain finite sets are considered.

Combination

A combination is an act of choosing items from a group, such that (not like permutation) the order of choice does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the union of n things taken k at a time without repetition  In combination you can select the items in any order. To those combinations in which re-occurrence is allowed, the terms k-selection or k-combination with replication are frequently used.

Permutation Formula

In permutation r things are selected from a set of n things without any replacement. In this order of selection matter.

nPr = (n!) / (n – r)!

n = set size, the total number of items in the set 

r = subset size, the number of items to be selected from the set

Combination Formula

 In combination r things are selected from a set of n things and where the order of selection does not matter.

^nC_r = \frac{^nP_r}{r!}=\frac{n!}{r!(n-r)!}

Here, n = Number of items in set

r = Number of items selected from the set 

How many committees can be made from 15 people that have at least 2 people in each committee

Solution:

Since order doesn’t matter 

Use the combination formula, n! ⁄ r!(n – r)!

Explanation:

n = 15, the total number of people

r = 2, the number of people being picked

Enter n and r into the formula:

= 15! ⁄ 2!(15 – 2)!

= 15! ⁄ 2!(13)!

= 150

So, there are 150 different committees that can be formed.

Similar Problems

Question 1: How many different 2 people committees can be formed from a group of 24 people?

Solution:

Since order doesn’t matter,

Use the combination formula, nCr = n! ⁄ r!(n – r)!

Explanation:

n = 24, the total number of people,

r = 2, the number of people being picked

Enter n and r into the formula:

= 24! ⁄ 2!(24 – 2)!

= 24! ⁄ 2!(22)!

= 276

So, there are 276 different committees that can be formed.

Question 2: How many different 4 people committees can be formed from a group of 25 people?

Solution:

Since order does not matter,

Use the combination formula, nCr = n! ⁄ r!(n – r)!

Explanation:

n = 25, the total number of people,

r = 4, the number of people being picked

Enter n and r into the formula:

= 25! ⁄ 4!(25 – 4)!

= 25! ⁄ 4!(21)!

= 12650

So, there are 12650 different committees that can be formed.

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