Problem 1 :
Determine the number of permutations of the letters of the word SIMPLE if all are taken at a time?
Solution :
Number of letters in the word "SIMPLE" = 6
All are unique letters.
Number of permutation = ^{6}P_{6 }= 6!
= 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1
= 720
Hence total number of permutation is 720.
Problem 2 :
A test consists of 10 multiple choice questions. In how many ways can the test be answered if
(i) Each question has four choices?
(ii) The first four questions have three choices and the remaining have five choices?
(iii) Question number n has n + 1 choices?
Solution :
(i) Each question has four choices?
Number of ways to answer 1^{st} question = 4
Number of ways to answer 2^{nd} question = 4
Number of ways to answer 3^{rd} question = 4
............................
Number of ways = 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4
= 4^{10}
Hence the total number of ways = 4^{10}
(ii) The first four questions have three choices and the remaining have five choices?
Number of ways to answer 1^{st} question = 3
Number of ways to answer 2^{nd} question = 3
Number of ways to answer 3^{rd} question = 3
Number of ways to answer 4^{th} question = 3
Number of ways to answer 5^{th} question = 5
Number of ways to answer 6^{th} question = 5
..............................
Number of ways = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5
= 3^{4}⋅ 5^{6}
Hence the total number of ways is 3^{4}⋅ 5^{6}.
(iii) Question number n has n + 1 choices?
Number of ways to answer 1^{st} question = 2
Number of ways to answer 2^{nd} question = 3
Number of ways to answer 3^{rd} question = 4
Number of ways to answer 4^{th} question = 5
...................
Number of ways to answer 10^{th} question = 11
Number of ways = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 ⋅ 8 ⋅ 9 ⋅ 10 ⋅ 11
= 11!
Hence the total number of ways is 11!.
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