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?
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 4C2 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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