in kitchen there are 7 cups 4 paper plates and 7 spoons

Question

How many ways can you select a cup, paper plate and a spoon

How many ways can you select 3 cups, 2 paper plates and 4 spoons

Answer 1

There are (7 choose 1) = 7 ways you can choose 1 cup, (4 choose 1) = 4 ways you can choose one plate, and (7 choose 1) = 7 ways you can choose one spoon.

 

That means there are 7×4×7 = 196 ways to choose one of each

 

 

 

Ways to choose 3 cups:

 

(7 choose 3) = 7!/(3!(7-3)!) = (7×6×5×4×3×2×1)/(3×2×1(4!)) = (7×6×5×4)/(4×3×2×1) = (7×6×5)/(3×2) = (7×6×5)/6 = 7×5 = 35 ways

 

Ways to choose 2 plates:

 

(4 choose 2) = 4!/(2!(4-2)!) = (4×3×2×1)/(2×1(2!)) = (4×3)/(2×1) = 12/2 = 6 ways

 

Ways to choose 4 spoons:

 

(7 choose 4) = 7!/(4!(7-4)!) = 7!/(3!×4!) = 7!/(3!(7-3)!) = (7 choose 3) = 35 ways, from above

 

 

So, there are 35×4×35 = 4900 ways to choose 3 cups, 2 plates, and 4 spoons