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in Combinatorics and Mathematical Induction by (46.9k points)
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How many strings can be formed using the letters of the word LOTUS if the word

(i) either starts with L or ends with S?

(ii) neither starts with L nor ends with S?

1 Answer

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Best answer

(i) To find the number of words starting with L 

Number of letters in LOTUS = 5 when the first letter is L it can be filled in 1 way only. So the remaining 4 letters can be arranged in 4! =24 ways = n(A). 

When the last letter is S it can be filled in the 1 way and the remaining 4 letters can be arranged is 4! = 24 ways = n(B)

Now the number of words starting with L and ending with S is OTU S

(1) (1) 3! = 6 = n(A ∩ B) 

Now n(A ∪ B) = n(A) + n(B) – n(A ∩ B) 

= 24 + 24 – 6 = 42 

Now, neither words starts with L nor ends with S = 42

(ii) Number of letters of the word LOTUS = 5. 

They can be arranged in 5 ! = 120 ways

Number of words starting with L and ending with S = 42 

So the number of words neither starts with L nor ends with S = 120 – 42 = 78

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