Question 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?
Solution :
Case 1 :
Number of words starts with L,
L ___ ___ ___ ___
1^{st} 2^{nd} 3^{rd} 4^{th} 5^{th}
1^{st} dash --> We have 1 option (L)
2^{nd} dash --> We have 4 options
3^{rd} dash --> We have 3 options
4^{rd} dash --> We have 2 options
5^{rd} dash --> We have 1 option
= 1 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1
= 24 words
Case 2 :
Number of words ends with S,
___ ___ ___ ___ S
1^{st} 2^{nd} 3^{rd} 4^{th} 5^{th}
5^{rd} dash --> We have 1 option (S)
1^{st} dash --> We have 4 options
2^{nd} dash --> We have 3 options
3^{rd} dash --> We have 2 options
4^{rd} dash --> We have 1 options
= 4 ⋅ 3 ⋅ 2 ⋅ 1 ⋅ 1
= 24 words
Case 3 :
Number of words starts with L and ends with S.
L ⋅ 3 ⋅ 2 ⋅ 1 ⋅ S
= 6
Either starts with L or ends with S = 24 + 24 - 6
= 42 words
Total number of words formed without restriction = 5!
= 120 words
Number of words starts with L nor S
= Total number of words - Number of words either starts with L or ends with S
= 120 - 42
= 78 words
Question 3 :
(i) Count the total number of ways of answering 6 objective type questions, each question having 4 choices.
(ii) In how many ways 10 pigeons can be placed in 3 different pigeon holes ?
(iii) Find the number of ways of distributing 12 distinct prizes to 10 students?
Solution :
(i) Each question is having 4 options.There are 4 ways to answer each question.
1^{st} question = 4 ways
2^{nd} question = 4 ways .........
Total number of ways = 4^{6}
(ii) In how many ways 10 pigeons can be placed in 3 different pigeon holes ?
Note : If n different objects are to be placed in m places, then the number of ways of placing is m^{n}.
10 - Number of pigeons (different things)
3 - Number of places (pigeon holes)
Applying the formula m^{n},
Total number of ways = 3^{10}
(iii) 12 - Number of prizes (different things)
10 - Number of place (students)
Applying the formula m^{n},
Total number of ways = 10^{12}
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