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Question

Consider the word ′′SUCCESS′′, then the total number of possible words

A
when all the letters are taken at a time is 420
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B
when All Ss are not together is 360
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C
when no two Ss are together is 180
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D
taking 4 letters at a time is 92
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Solution

The correct option is B when All Ss are not together is 360
We have, 2Cs,1E,3Ss,1U
Total permutations when all the letters are used =7!3!2!=420

When All Ss are not together.
Total ways when All Ss are together =5!2!=60
Required number of ways =42060=360

When no two Ss are together.
Arrange other words in 4!=24 ways.
Now the Ss can be arrnged between the gaps in 5C3=10 ways
Required number of ways =24×10=240

Possible words taking 4 letters at a time.

Case 1: When all the four letters are distinct:-
4P4=4!=24

Case 2: When only two letters are repeated:-
the repeated letter can be selected from C,S i.e. 2C1 and the remaining two letters can be selected from 3 i.e. 3C2. After selecting these 4 letters we have to arrange them while keep in mind that 2 of them are same.
Number of ways
=(2C1× 3C2)4!2!
=72

Case 3: When one is repeated twice and another one is repeated twice:-
the repeating letter can be selected as C,S i.e. 2C2
Number of ways
= 2C2×4!2!2!
=6

Case 4: When one is repeated thrice and other is once:-
Number of ways:
(1C1× 3C1)4!3!
=1×3×4
=12
Total words =24+72+6+12=114

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