How many strings can be formed using the letters of the word LOTUS if

1.	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?
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 L 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

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?

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 L 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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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?