Anna has five circular discs that are all of different sizes. She wants to build a tower usingthree discs where a smaller disc always has to lie on top of a bigger disc.How many ways are there for Anna to build the tower?(A) 5 (B) 6 (C) 8 (D) 10 (E) 15
Question
Anna has five circular discs that are all of different sizes. She wants to build a tower usingthree discs where a smaller disc always has to lie on top of a bigger disc.How many ways are there for Anna to build the tower?(A) 5 (B) 6 (C) 8 (D) 10 (E) 15
Solution
This problem can be solved using the concept of combinations in mathematics.
Anna has 5 discs and she wants to select 3 of them to make a tower. The order in which she selects the discs matters because a smaller disc has to lie on top of a bigger disc.
The number of ways to select 3 discs out of 5 (order matters) is given by the formula for permutations:
P(n, r) = n! / (n-r)!
where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.
So, in this case, n = 5 (total discs) and r = 3 (discs to choose for the tower).
P(5, 3) = 5! / (5-3)!
= 120 / 2
= 60
However, for each selection of 3 discs, there is only one way to arrange them (smallest on top, largest on bottom). So, the total number of ways to build the tower is 60 / 3! = 60 / 6 = 10.
So, the answer is (D) 10.
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