# The number of ways of selecting 6 objects from a bag that has n different objects is 28. What is n?

justaguide | College Teacher | (Level 2) Distinguished Educator

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The number of ways of selecting r objects from a set of n different objects is given by `C(n, r) = (n!)/(r!*(n - r)!)` .

The number of ways of selecting 6 objects from a bag that has n different objects is 28. It is assumed here that the order in which they are selected is not important.

To determine n, solve the equation `(n!)/(6!*(n - 6)!) = 28`

`(n!)/(6!*(n - 6)!) = 28`

=> n*(n-1)*(n-2) ... (n-5) = 28*6!

=> n*(n-1)*(n-2) ... (n-5) = 28*720

=> n*(n-1)*(n-2) ... (n-5) = 20160 ...(1)

The product of 6 consecutive numbers is 20160

Solving (1) is going to give 6 roots and is very difficult to do manually. Instead, use a trial and error method to determine n. It is known that n is an integer and greater than 6.

If n = 7, C(7, 6) = 7

If n = 8, C(8, 6) = 28

This gives the number of objects in the bag as 8.