This sample GMAT Math question is a Permutation problem solving question. The concept tested is to find the number of ways of selecting an object from identical objects. A GMAT 650 level, interesting permutation combination question.

Question 11: There are 4 identical pens and 7 identical books. In how many ways can a person select at least one object from this set?

- 12
- (2
^{4}– 1)(2^{7}– 1) - 11
- 2
^{11}- 1 - 39

@ INR

What if the 4 pens were distinct?

If the pens are distinct, the number of ways of selecting one pen is 4 ways.

If the pens are identical, the number of ways of selecting a pen is 1 way, as all of them are the same.

If the pens are distinct, the number of ways of selecting two pens is 4C_{2} ways.

If the pens are identical, the number of ways of selecting two pens is 1.

This information is key to how we approach the problem.

We can select anything from 0 to 4 pens.

Therefore, 0 or 1 or 2 or 3 or 4 pens can be selected in 5 ways.

Similarly, we can select from 0 to 7 books.

Hence, 0 or 1 or 2 or 3 or 4 or 5 or 6 or 7 pens can be selected in 8 ways.

The number of ways of selecting none or all of the available objects is 8 × 5 = 40 ways.

However, we have to select at least one object from these two.

Hence, the possibility of selecting 0 books and 0 pens has to be subtracted from the overall possibilities.

∴ 40 - 1 = **39 ways**.

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