An interesting GMAT problem solving practice question in permutation combination. The concept covered in this question is selecting one or more objects from a set of object, all of which are not distinct. The objects not being distinct is what makes this question an interesting one. This is another question that you can use as a template to solve questions that comprise elements that are not distinct.

### Question

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

a. 12

b. (2^{4} – 1)(2^{7} -1)

c. 11

d. 2^{11} – 1

e. 39

### Correct Answer

Choice E. 39 ways.

### Explanatory Answer

**Demystifying number of ways of selecting when we are presented with identical objects**

The 4 pens stated in the question are identical. Let us understand how that makes a difference.

Had these pens been distinct, the number of ways of selecting one pen out of these 4 would have been 4c1 or 4 ways. For e.g., if we name the pens A, B, C, and D – selecting A is different from selecting B.

But because these pens are identical, the number of ways of selecting one pen out of the 4 is just 1 way. It does not matter which one you picked it will appear the same as picking one of the others.

Let us extend the reasoning to selecting two pens. Had these pens been distinct, the number of ways of selecting two pens out of 4 is 4c2 = 6 ways. For e.g., if we name the 4 distinct pens A, B, C, and D – selecting AB is different from selecting BD.

However, because these pens are identical, the number of ways of selecting two pens out of four is also just 1 ways. You take any two out of the four, it is going to appear the same.

#### What do we have in this question?

4 identical pens and 7 identical books. We have to pick at least one object.

A person can select none or up to 4 identical pens in 5 ways (0 or 1 or 2 or 3 or 4 pens).

A person can select none or up to 7 identical books in 8 ways (0 or 1 or 2 or .. 7 books).

So, one has 5 ways of selecting pens and 8 ways of selecting books. (These include the option of not selecting any pen and any book)

So, a person can select none or all of the objects in 5 * 8 = 40 ways.

However, one of these 40 ways includes the scenario that neither a pen nor book would have been selected. We need to select at least one object. So, let us eliminate that one possibility.

Therefore, number of ways of selecting at least one object from 4 identical pens and 7 identical books = 40 – 1 = 39.

Note: Had the objects been distinct, we could select none or all objects in 2^{11} ways.

**Why so?** Because each of the objects has 2 choices – being selected or not being selected.

11 objects have 2^{11} outcomes.

However, if we have to select at least one object, we have to eliminate the only outcome in which we select none of the objects.

Had the objects been distinct, the answer would have been 2^{11} – 1.

hsb says

39

total ways = (7+1)(4+1)= 40

but it'll include the case where neither a pen nor a book is selected so subtract 1

hsb says

39 is the answer ??

total ways = (7+1)(4+1)-1 = 40-1= 39

Dhananjay Kumar says

Dear Sir,

As questions ask at least one object, how can we select at least one from each article. Total article 4+7 = 11 and at least one should be selected in 11 ways. And if at least one from each article then 4*7.

It is not asking for at least each of both. I am confused. As in different article example if we select at least one object, the ans 2^11 – 1. And if at least one from each article the ans is (2^7-1) * (2^4-1)

What is the difference

K S Baskar says

Hello Dhananjay!

Why is it not 4 + 7 = 11?Sounds the most obvious answer.Let us take 2 examples to understand the logic behind the answer

Example 1: If we had 11 identical objects – say all 11 of them are pens, then the number of ways of selecting at least one object will be 11.Example 2: Now, if these 11 objects comprised 4 identical pens and 7 identical books, why is it not 11?Let us compare examples 1 and 2.

In how many ways can we take 2 objects out of 11 in example 1. 1 way. All 11 are identical. You will not notice any difference between which 2 you took out of the 11.

In how many ways can we take 2 objects out of 11 in example 2. Example 2 has 4 identical pens and 7 identical books.

1. The 2 objects could both be pens. That is 1 way.

2. The 2 objects could both be books. That is 1 way.

3. The 2 objects could be one pen and one book. That is 1 way.

So, now we have 3 ways of selecting 2 objects out of 11 in example 2 as against just 1 way when all 11 were identical objects. To help consolidate your learning, list down all the possibilities from selecting 1 object to selecting 11 objects in example 2. That will help you appreciate the difference clearly.

Even for 1 object in example 2, we will have 2 ways. The object could be a pen or a book.

This accounts for the difference in approach that we employ to solve this question.