Statistics

A student chooses 4 books at random from a reading list of 11 books.

Step-by-step solution with explanation

Final Answer

different combinations of books

Step-by-step solution

1

Identify the type of counting problem

The student is just choosing books, not arranging them. When order doesn't matter, we use combinations, not permutations.
2

Write the combination formula

Here n is the total number of items to choose from, and r is how many we pick. The factorial signs mean we multiply descending whole numbers.
3

Plug in the values

We have 11 books total and we are choosing 4. We substitute n = 11 and r = 4 into the formula.
4

Cancel the large factorials

The 7! on top and bottom cancel out, leaving only the top 4 factors in the numerator. The denominator is 4! = 4 × 3 × 2 × 1 = 24.
5

Divide to get the final count

Dividing 7920 by 24 gives us 330. This is the total number of different groups of 4 books the student could choose.

Understanding this problem

Learning Insight

Combinations count the number of ways to select a group when order doesn't matter. Dividing by r! removes all the duplicate groups that would appear if we had counted arrangements. That's why combinations are always smaller than permutations for the same n and r.

Quick Tip

To save time, only expand as many factors in n! as there are in r. Here r = 4, so just take the top 4 factors of 11! (11×10×9×8) and divide by 4!. The rest cancels automatically.

Common Mistake

Students often use the permutation formula P(11,4) instead of C(11,4), which gives 7920 — but that counts every arrangement of the same 4 books as different, which is wrong when we only care about which books are chosen.