Answer:
(a) 720
(b) 180
Step-by-step explanation:
We are given 10 numbers (1,2,3,4,.....,10).
(a) 3 unique numbers are to be chosen by Alice, Bob and Carrie.
Number of ways to choose the 1st number = 10
Number of ways to choose the 2nd number = 9 (1 number chosen earlier can't be chosen again)
Number of ways to choose the 3rd number = 8 (2 numbers chosen earlier can't be chosen again)
Total number of ways of choosing 3 unique numbers = [tex]10 \times 9 \times 8 \Rightarrow 720[/tex]
(b)
Number of ways of choosing 1st number = 10 (no restrictions)
Number of ways of choosing 2nd number = 9 (1 number chosen earlier can't be chosen again)
Now, restriction on the 3rd number to be chosen is that one of the two already chosen number should be chosen again.
So, number of ways to choose 3rd number = 2
the number of ways they can choose their numbers if they have to choose exactly two different numbers among all three people =
[tex]10 \times 9 \times 2 \Rightarrow 180[/tex]
Hence, the answers are:
(a) 720
(b) 180
