(a) The producer of a play requires a total cast of 5, of which 3 are actors and 2 are actresses. He
auditions 5 actors and 4 actresses for the cast. Find the total number of ways in which the cast can
be obtained.
[3]
(b) Find how many different odd 4-digit numbers less than 4000 can be made from the digits
1, 2, 3, 4, 5, 6, 7 if no digit may be repeated.
[3]
Answers
Given : The producer of a play requires a total cast of 5, of which 3 are actors and 2 are actresses. He auditions 5 actors and 4 actresses for the cast.
To Find : the total number of ways in which the cast can be obtained.
Solution:
Actors for Auditions = 5
Actors to be selected = 3
Ways of Select 3 actors out of 5 = ⁵C₃ = 10
Actresses for Auditions = 4
Actresses to be selected = 2
Ways of Select 2 Actresses out of 4 = ⁴C₂ = 6
total number of ways in which the cast can be obtained. = 10 x 6
= 60
odd 4-digit numbers less than 4000 can be made from the digits
1, 2, 3, 4, 5, 6, 7 if no digit may be repeated.
1st Digit can be 1 , 2 , 3
4th Digit can be 1 , 3 , 5 , 7 ( odd numbers )
1st Digit (1 or 3 ) - 2 ways then 4th Digit 3 ways
remaining 2 digits out of 5 digits can be selected in 5*4 = 20 Ways
= 2 * 3 * 20 = 120
1st Digit 2 ( 1 way) then 4th Digit 4 ways
remaining 2 digits out of 5 digits can be selected in 5*4 = 20 Ways
= 1 * 4 * 20 =80
Total = 120 + 80 = 200
200 different odd 4-digit numbers less than 4000 can be made from the digits 1, 2, 3, 4, 5, 6, 7
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