Combination Calculation for Selecting Books: A Detailed Guide

When dealing with the selection of books from different categories—math, science, and literature—understanding the principle of combinations becomes crucial. This article delves into the steps and calculations needed to determine the number of ways to choose two books from each category. By applying the combination formula, we can accurately find the total number of possible selections.

Understanding the Combination Formula

Combinations are a fundamental concept in mathematics that allow us to determine the number of ways a set of items can be selected without regard to the order. The combination formula is given by:

Combination (n, r) n! / [r!(n-r)!]

Where n is the total number of items, and r is the number of items to be selected. This formula helps us avoid counting permutations, which would have considered the order in which items are selected.

Calculating Combinations for Each Category

Let's apply the combination formula to each category of books:

Math Books

We need to select 2 books from a set of 7:

Combination (7, 2) 7! / [2!(7-2)!] (7 * 6) / (2 * 1) 21

Science Books

We need to select 2 books from a set of 9:

Combination (9, 2) 9! / [2!(9-2)!] (9 * 8) / (2 * 1) 36

Literature Books

We need to select 2 books from a set of 5:

Combination (5, 2) 5! / [2!(5-2)!] (5 * 4) / (2 * 1) 10

These combinations give us the number of ways to choose 2 books from each respective category independently.

Total Number of Ways to Select

Since the selections are independent, we can find the total number of ways to choose the books by multiplying the number of combinations from each category:

Combination (7, 2) * Combination (9, 2) * Combination (5, 2) 21 * 36 * 10 7560

Therefore, the total number of ways a student can select two books from each set is 7560.

Alternative Methods of Calculation

Alternatively, we can calculate the combinations for each category directly:

(9 * 8) / 2 * (7 * 6) / 2 * (5 * 4) / 2

Breaking down the multiplication:

9 * 8 72, 7 * 6 42, 5 * 4 20

Then:

72 / 2 * 42 / 2 * 20 / 2 36 * 21 * 10 7560

This confirms the same result, illustrating the power of the combination formula and its application in real-world scenarios.

Conclusion

By understanding and applying the combination formula, we can accurately determine the number of ways to select books from different categories. This is particularly useful in scenarios involving multiple categories and independent selections. The total number of combinations for selecting two books from each category is 7560, as demonstrated through various methods of calculation.