Restricted combination is an important topic in permutations and combinations. In this article, we will learn the formula for the restricted combination along with solved examples. This will definitely help students to easily crack problems related to restricted combination.
Combination deals with ways of selection. nCr means total ways of selection of r things out of n things. Let us discuss an example of a restricted combination. Consider that we need to select r players out of n players to form a team. There are 2 excellent players who are always particularly selected. So, we need to select only (r-2) players from (n-2) players. The formula for various cases of restricted combination is given below.
Formula for Restricted Combination
1) Number of ways of selecting r things out of n things = nCr
2) Number of ways of selection from n different things, taken r at a time, when p particular things are always selected = n-pCr-p.
3) Number of ways of selection from n different things, taken r at a time, when p particular things are always to be rejected = n-pCr
4) Number of ways of selection of r different things from n things, where k things are always selected, and p things are always rejected = n-k-pCr-k
Also, Read:
Permutations and Combinations Solved Problems
Solved Examples
Example1:
Out of 16 players, 11 are to be selected for a cricket team. If the wicketkeeper and captain are to be always included in each selection, in how many ways can this be done?
1) 14C9
2) 16C11
3) 14C11
4) none of these
Solution:
Given n = 16
r = 11
No. of players always included, p = 2
So, the number of ways = n-pCr-p
= 14C9
hence, option 1 is the answer.
Example 2:
A group of students consists of 5 boys and n girls. If the number of ways in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to:
1) 24
2) 25
3) 26
4) 28
Solution:
Given the number of boys = 5
Number of girls = n
We can select 1 boy and 2 girls or 2 boys and 1 girl.
Number of ways of selecting 3 students under the given criteria => 5C1 .nC2 + 5C2 .nC1 = 1750
=> 5×n(n-1)/2 + (5×4/1×2)n = 1750
=> 5×n(n-1)/2 + 10n = 1750
=> 5n2 + 15n – 350 = 0
=> n2 + 3n – 700 = 0
=> (n + 28)(n – 25) = 0
=> n = -28, n = 25
So, n = 25 (n cannot be negative)
Hence, option 2 is the answer.
Example 3:
Find the number of ways of selecting of 11 players out of 16, if the wicketkeeper and captain are always included in each selection, and 1 player is injured.
1) 14C9
2) 13C9
3) 13C11
4) none of these
Solution:
Total number of players, n = 16
Number of players to be selected, r = 11
Number of players who are always included, k = 2
Number of players rejected, p = 1 (injured player)
Number of ways of selection = n-k-pCr-k
= 16-2-1C11-2
= 13C9
Hence, option 2 is the answer.
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Frequently Asked Questions
What is meant by combination?
A combination is an arrangement of objects. Here, the order of selection does not matter.
Give the number of ways of selecting from n different things, taken r at a time, when p particular things are always to be rejected.
The number of ways of selection from n different things, taken r at a time, when p particular things are always to be rejected, is given by the equation n-pCr.
In how many ways can we select r things from n identical things?
The number of ways of selecting r things from n identical things is 1.
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