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Question
How many strings can be formed using the letters of the word LOTUS if the word either starts with L or ends with S?
Solution
Either starts with L or ends with S
| L |
The first box is filled with the letter L.
The second box can be filled with the remaining letters O, T, U, S in 4 ways.
The third box can be filled with the remaining letters excluding L and the letter placed in box 2 in 3 ways.
The fourth box can be filled with the remaining letters excluding L and the letters placed in a box – 2 and box – 3 in 2 ways.
The fifth box can be filled with the remaining one letter excluding L and the letters placed in a box – 2 and box – 3, box – 4 in 1 way.
Therefore, by fundamental principle of multiplication
The number of words start with L is = 1 × 4 × 3 × 2 × 1 = 24
| S |
Since the word ends with S, the fifth box can be filled in one way with the letter S.
The remaining four boxes can be filled 4 × 3 × 2 × 1 way.
Therefore, the number of words ending with S = 4 × 3 × 2 × 1 × 1 = 24
Number of words starting with L and ends with S:
The first box can be filled with L in one way
The Fifth box can be filled with S in one way second box,
The Third box and fourth box can be filled in 3 × 2 × 1 ways with the remaining letters O, T, U.
∴ Number of words starting with L and ends with S = 1 × 3 × 2 × 1 × 1 = 6
Therefore, by fundamental principle of addition
Number of words either starts with L or ends with S = 24 + 24 – 6 = 48 – 6 = 42
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