Question

A catering service offers 4 appetizers, 10 main courses, and 9 desserts. A banquet chairperson is to select 3 appetizers, 8 main courses, and 3 desserts, for a banquet. In how many ways can this be done?

Answer 1

Answer: 15120

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Work Shown:

There are 4 appetizers total. We want to select 3 of them. Order doesn't matter. 
Use the nCr formula with n = 4 and r = 3
n C r = (n!)/(r!*(n-r)!)
4 C 3 = (4!)/(3!*(4-3)!)
4 C 3 = (4!)/(3!*1!)
4 C 3 = (4*3!)/(3!*1!)
4 C 3 = (4)/(1!)
4 C 3 = (4)/(1)
4 C 3 = 4/1
4 C 3 = 4
There are 4 ways to select just the appetizers
Call this value M = 4 (we'll use it later)
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There are 10 main courses total. We want to select 8 of them. Order doesn't matter. 
Use the nCr formula with n = 10 and r = 8
n C r = (n!)/(r!*(n-r)!)
10 C 8 = (10!)/(8!*(10-8)!)
10 C 8 = (10!)/(8!*2!)
10 C 8 = (10*9*8!)/(8!*2!)
10 C 8 = (10*9)/(2!)
10 C 8 = (10*9)/(2*1)
10 C 8 = 90/2
10 C 8 = 45
There are 45 ways to select just the main courses
Call this value N = 45 (we'll use it later)
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There are 9 desserts total. We want to select 3 of them. Order doesn't matter. 
Use the nCr formula with n =9 and r = 3
n C r = (n!)/(r!*(n-r)!)
9 C 3 = (9!)/(3!*(9-3)!)
9 C 3 = (9!)/(3!*6!)
9 C 3 = (9*8*7*6!)/(3!*6!)
9 C 3 = (9*8*7)/(3!)
9 C 3 = (9*8*7)/(3*2*1)
9 C 3 = 504/6
9 C 3 = 84 
There are 84 ways to select just the desserts
Call this value P = 84 (we'll use it later)
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Multiply the results from earlier:
M*N*P = 4*45*84 = 15120 which is the final result.