# 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. [duplicate]

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?

## marked as duplicate by Henning Makholm combinatorics StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 7 at 12:53

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 $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$
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?