Thread: Set theory problem

Need help solving the following. I cannot find enough information in the text to begin solving the problem.

1/4 of the five element subsets of (1,2,3...n) contain the element 7, determine n. n >5.

There are subsets of {1, 2, 3, ..., n} that contain 5 numbers. 1/4 of that would be and "1/4 of the five element subsets of (1,2,3...n) contain the element 7" implies that must be an integer.

I have been struggling with the equation for hours trying to solve for n without success. Is there enough information given to solve this problem?

Yes, there is enough information given to solve for .

HallsofIvy has told you that 1/4 of the number of subsets containing 5 elements is:



Now, to find the number of subsets of cardinality 5 containing 7 as an element, we may use the fundamental counting principle to state:



Hence, we have:



Now, you just need to solve for .

I think the equation I need to solve for n is

1/4*(n,5)=(n-1,4)

but I have no idea how to solve this equation

This equation is the same as the last equation in MarkFL2 post. Multiply both sides by (n-5)!. Then divide both sides by (n-1)! (remember that n!/(n-1)! = n). Then multiply oth sides by 4!.

Originally Posted by wattskickin

I think the equation I need to solve for n is
1/4*(n,5)=(n-1,4) but I have no idea how to solve this equation

You have been told exactly what to do.

Solve