A tea shoppe serves 12 different flavors of tea. 4 customers each order a cup...

Montenovofe

Montenovofe

Answered

2022-07-12

A tea shoppe serves 12 different flavors of tea. 4 customers each order a cup of tea. Assuming that each customer's choice is completely random and is independent of any other customer's selection:
1) what is the probability that 3 different flavors are selected by the 4 customers (e.g:: if letters A-L represent flavors, AABC)?
2) what is the probability that only 2 different flavors are selected by these 4 customers (e,g: AABB or AAAB)? 3) what is the probability that 2 students choose 1 flavor and the other 2 choose another flavor (e.g: AABB)?

Answer & Explanation

lofoptiformfp

lofoptiformfp

Expert

2022-07-13Added 16 answers

If the four customers select three different flavors, then two of the four customers choose one of the 12 flavors, a third customer chooses one of the remaining 11 flavors, and the fourth customer chooses one of the remaining 10 flavors, which can be done in
( 4 2 ) P ( 12 , 3 ) = 7920 ways.
The number of ways three of the four customers can choose one of the 12 flavors and the fourth chooses one of the remaining 11 flavors is
( 4 3 ) P ( 12 , 2 ) = 528
The number of ways one pair selects one of the 12 available flavors while the other pair chooses one of the remaining 11 flavors is
1 2 ( 4 2 ) P ( 12 , 2 ) = 396
The factor of 1/2 is necessary since the same selection results when the first pair chooses flavor A then the second pair chooses flavor B as when the second pair select flavor B then the first pair chooses flavor A. Thus, the number of ways that the customers can select two different flavors is
( 4 3 ) P ( 12 , 2 ) + 1 2 ( 4 2 ) P ( 12 , 2 ) = 528 + 396 = 924
Hence, the total number of ways of selecting either two or three different flavors is
924 + 7920 = 8844

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