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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 nC r.

 

Description: 59006.png 

Properties of Combinations

  • ƒ Description: 59013.png 
  • ƒ Description: 59023.png 
  • ƒ Description: 59029.png 
  • ƒ Description: 59040.png 
  • ƒ Description: 59046.png 
  • ƒ Description: 59058.png 

Example
If nC7 = nC3, then find the value of n.
Solution
Given nC7 = nC3
Comparing with the property Description: 59064.png
Clearly, p = 7 and q = 3 are not equal. Thus, p + q = n.
Description: 59071.png 

 

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 15C11 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 = 4C2 × 5C2 = 60
3 girls and 1 boy = 4C3 × 5C1 = 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 nC1 + nC2 nC3 + ... + nCn = 2n  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 210 – 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 Description: 59078.png

 

Case 3: The number of combinations of n items taken r at a time in which p particular items never occur is (n – p)Cr.
 

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 Description: 59088.png

 

Example
Find the number of ways in which 3 players can be selected out of 6 players.
Solution
Required no. of ways Description: 59094.png

The notion of Independence in Combination

The combinations of selecting r1 things from a set having n1 objects and r2 things from a set having n2 objects, where combination of r1 things are independent of r2 things is given by Description: 59101.png

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 Description: 59107.png 
The number of ways in which (m + n) things can be divided into two groups containing m and n objects respectively is Description: 59113.png
Similarly, (m + n + p) things can be divided into 3 groups containing mn and p things respectively in
Description: 59123.png





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