A catering service offers 12 appetizers, 9 main courses, and 7 desserts. A banquet chairperson is to select 8 appetizers, 8 main courses, and 6 desserts for a banquet.

In how many ways can this be done?

12 − a pp e t i zers , 9 − main co u rses , 7 d esser t s se l ec t i o n : 8 a pp e t i zers , 8 main co u rses , 6 d esser t s a = t h e n u mb er o f se l ec t i o n o f a pp e t i zers : ( 8 12 ​ ) = 8 ! ⋅ ( 12 − 8 )! 12 ! ​ = 8 ! ⋅ 4 ⋅ 3 ⋅ 2 12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8 ! ​ = 11 ⋅ 5 ⋅ 9 = 495 c = t h e n u mb er o f se l ec t i o n o f main co u rses : ( 8 9 ​ ) = 8 ! ⋅ ( 9 − 8 )! 9 ! ​ = 8 ! ⋅ 1 9 ⋅ 8 ! ​ = 9
d = t h e n u mb er o f se l ec t i o n o f d esser t s : ( 6 7 ​ ) = 6 ! ⋅ ( 7 − 6 )! 7 ! ​ = 6 ! ⋅ 1 7 ⋅ 6 ! ​ = 7 t h e n u mb er o f se l ec t i o n se t s t h e ban q u e t : a ⋅ c ⋅ d = 495 ⋅ 9 ⋅ 7 = 31185

Answered by kate200468 | 2024-06-10

[(12! / (8! 4!) ] [9! / (8!*1!) ] * [ 7!/ (6!*1!)] = ( 12 * 11 * 10 * 9 / 4 * 3 * 2 * 1) * 9 * 7 = 45 * 11 * 9 * 7 = 31185 ways

Answered by crisforp | 2024-06-10

The total number of ways the banquet chairperson can select 8 appetizers from 12, 8 main courses from 9, and 6 desserts from 7 is 31,185. This was calculated using the combination formula and the Multiplication Principle of counting. Each selection event was calculated separately before multiplying the results for the final answer.
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Answered by kate200468 | 2024-12-23