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?

Answer 1

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