Combinations
The number of ways of selecting r things from given n things is called combination of n things by taking r things at a time. It is written as ^{n}C _{r}.
Properties of Combinations






Example
If ^{n}C_{7} = ^{n}C_{3}, then find the value of n.
Solution
Given ^{n}C_{7} = ^{n}C_{3}
Comparing with the property
Clearly, p = 7 and q = 3 are not equal. Thus, p + q = n.
Comparing with the property
Clearly, p = 7 and q = 3 are not equal. Thus, p + q = n.
Example
In how many ways can a cricket team of 11 be selected from a group of 15?
Solution
11 players must be selected from 15 players.
This can be done in ^{15}C_{11} ways = 1365.
This can be done in ^{15}C_{11} ways = 1365.
Example
There are 5 boys and 4 girls. Find the number of ways in which a dance team of 4 members can be formed from these, if the team must include at least 2 girls and at least 1 boy.
Solution
The team consisting of 4 members can be formed in the following ways:
2 girls and 2 boys = ^{4}C_{2} × ^{5}C_{2} = 60
3 girls and 1 boy = ^{4}C_{3} × ^{5}C_{1} = 20
Hence, the total number of selections is 60 + 20 = 80.
2 girls and 2 boys = ^{4}C_{2} × ^{5}C_{2} = 60
3 girls and 1 boy = ^{4}C_{3} × ^{5}C_{1} = 20
Hence, the total number of selections is 60 + 20 = 80.
Types of Combinations
Case 1: The number of ways of selecting one or more items out of n given items is ^{n}C_{1} + ^{n}C_{2 }+ ^{n}C_{3} + ... + ^{n}C_{n} = 2^{n} − 1.
Example
In how many ways can 1 or more students in a class of 10 be selected to give a seminar?
Solution
Here, it means that either 1 student can be selected or 2 students can be selected or 3 students can be selected and so on. All 10 students may also be selected.
In this case, the total number of possible selections are 2^{10} – 1.
Case 2: The number of combinations of n items taken r at a time in which given p particular items will always occur is ^{(n – p)}C_{(r – p)}.
Here, p objects will always be selected. So, let’s take those objects first because their selection is certain. Now, the remaining objects we have are (n – p) from which (r – p) have to be selected. This can be done in ^{(n – p)}C_{(r – p)} ways.
Example
Find the number of ways in which 4 players can be selected from 8 players so that 2 particular players are always included.
Solution
Required no. of ways
Case 3: The number of combinations of n items taken r at a time in which p particular items never occur is ^{(n – p)}C_{r}.
Example
Find the number of ways a volley ball team of 6 can be chosen out of a batch of 10 players so that a particular player is excluded.
Solution
Required no. of ways
Example
Find the number of ways in which 3 players can be selected out of 6 players.
Solution
Required no. of ways
The notion of Independence in Combination
The combinations of selecting r_{1} things from a set having n_{1} objects and r_{2} things from a set having n_{2} objects, where combination of r_{1} things are independent of r_{2} things is given by
Example
Find the number of ways 1 vowel and 2 consonants can be chosen from the letters of the word “VIKRAM”.
Solution
In the word VIKRAM, number of vowels =2, no. of consonants = 4
Therefore, required no. of ways
The number of ways in which (m + n) things can be divided into two groups containing m and n objects respectively is
Similarly, (m + n + p) things can be divided into 3 groups containing m, n and p things respectively in
Therefore, required no. of ways
The number of ways in which (m + n) things can be divided into two groups containing m and n objects respectively is
Similarly, (m + n + p) things can be divided into 3 groups containing m, n and p things respectively in