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Q.

In how many ways 4 letters can be selected from the letters of the word INEFFECTIVE?

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answer is 89.

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Detailed Solution

INEFFECTIVE contains 11 letters: EEE, FF, II, C, T, N, V. We will make following cases to select 4letters.

Case 1: 3 alike and 1 distinct

3 alike letters can be selected from 1 set of 3 alike letters (EEE) in 1 way.

The number of ways to select 3 alike letters =1

The number of ways to select 1 distinct letters =6

Total ways =6×1=6

Case 2: 2 alike and 2 alike

'2 alike and 2 alike' means we have to select 2 groups of 2 alike letters (EE, FE, II) in  C2  3 ways.

 The number of ways to select '2  alike and 2 alike' letters =C2=3  3

Case 3: 2 alike and 2 distinct

1 group of 2 alike letters can be selected from 3 sets of 2 alike letters (EE, FE, II) is  C1  3 ways.

2 distinct letters can be selected from 6 distinct letters (C,T,N,V, remaining 2 sets of two letters alike) in C2  6 ways.

The number of ways to select '2  alike and 2 distinct letters'

C1  3×C2=3×15=45  6

Case 4: AII distinct letters

All distinct letters can be selected from 7 distinct letters (I, E, F, N, C, T, V) in C4  7 ways.

 The number of ways to select all distinct letters =C4=35  7

Combining (1), (2), (3) and (4), we get,

Total number of ways to select 4 letters from the letter word 'INEFFECTIVE' =6+3+45+35=89

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In how many ways 4 letters can be selected from the letters of the word INEFFECTIVE?