GMAT Practice Question: There are 10 books on a shelf, of which 4 are paperbacks and 6 are hardbacks. How many possible...

Step 1

To find the number of ways to select 5 books with at least one paperback and one hardback, first count all possible selections, then subtract the cases where all books are of one type.

Step 2

Write the formula for the number of ways to choose 5 books from 10.

C(10,5) = \frac{10!}{5! \times 5!}

Step 3

Expand the factorials and cancel 5! in the numerator and denominator.

= \frac{10 \times 9 \times 8 \times 7 \times 6 \times \cancel{5!}}{\cancel{5!} \times 5 \times 4 \times 3 \times 2 \times 1}

Step 4

Write the remaining numerator and denominator after cancellation.

= \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1}

Step 5

Factorize numbers in the numerator and denominator to allow only identical factors to be cancelled.

= \frac{(5 \times 2) \times (3 \times 3) \times (4 \times 2) \times 7 \times (3 \times 2)}{5 \times 4 \times 3 \times 2 \times 1}

Step 6

Cancel identical factors in the numerator and denominator.

= \frac{\cancel{5} \times 2 \times \cancel{3} \times 3 \times \cancel{4} \times 2 \times 7 \times \cancel{3} \times 2}{\cancel{5} \times \cancel{4} \times \cancel{3} \times \cancel{2} \times 1}

Step 7

Multiply the remaining numbers to get the total number of ways.

= 2 \times 3 \times 2 \times 7 \times 3 = 252

Step 8

Subtract the selections that are all hardbacks or all paperbacks from the total to get those with at least one of each type.

252 - 6 - 0 = 246