Anna has five circular discs of different sizes. She wants to build a tower of four discs so that each disc in her tower is smaller than the disc immediately below it. How many different towers could Anna build?4591220
Question
Anna has five circular discs of different sizes. She wants to build a tower of four discs so that each disc in her tower is smaller than the disc immediately below it. How many different towers could Anna build?4591220
Solution
To solve this problem, we need to use the concept of combinations in mathematics.
A combination is a selection of items where order does not matter. In this case, Anna is selecting 4 discs out of 5 to make a tower, and the order in which she selects the discs does not matter because she can always rearrange them from largest to smallest.
The formula for combinations is:
C(n, k) = n! / [k!(n-k)!]
where:
- n is the total number of items,
- k is the number of items to choose,
- "!" denotes factorial, which is the product of all positive integers up to that number.
So in this case, n = 5 (the total number of discs) and k = 4 (the number of discs Anna wants to use in her tower).
Substituting these values into the formula gives:
C(5, 4) = 5! / [4!(5-4)!] = (54321) / [(4321)(1)] = 5
So, Anna can build 5 different towers with her discs.
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