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What is the number of subsets of $\{1,2,\ldots,10\}$ with three elements that contain at least one even number and at least one odd number?

I know if its either odd or even we can subtract $2^{10}$ with $2^5$, but how do you find the number of subsets with $3$ elements with at least one even number and at least one odd number?

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marked as duplicate by Henning Makholm combinatorics May 7 at 12:53

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  • $\begingroup$ Hint: use inclusion exclusion. $\endgroup$ – lulu May 7 at 10:26
  • $\begingroup$ Please read this tutorial on how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig May 7 at 10:38
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Note that $2^{10}$ is the number of subsets of any size of a set with $10$ elements. The number of subsets of size $k$ of a set with $n$ elements is $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$

Strategy: To find the number of three-element subsets that do not contain at least one even number and at least one odd number, we must subtract those that contain no even numbers or no odd numbers from the total number of three-element subsets. Notice that it is not possible for a three-element subset to contain neither even nor odd numbers.

There are $\binom{10}{3}$ ways to select a subset with three elements from a set with ten elements. How many of these contain no even numbers? no odd numbers?

Alternate Strategy: If a three-element subset contains at least one even number and at least one odd number, it either contains two even numbers and one odd number or it contains one even number and two odd numbers.

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