Combination Formula
The coach of the Wildcats now knows that he has to use the equation nCr = n!/r!(n-r)!, where n represents the number of items and r represents the number of items being chosen at a time. Using this equation, he must select two teams for each game from the eight teams in the district. So, the variable n would equal 8 and the variable r would equal 2. The equation would then look like 8 C 2 = 8!/2!(8-2)!.
To solve this equation, we would first need to perform (8-2) in the parenthesis, which would equal (8!/(2! x 6!). Next, we would expand 8!, 2! and 6!. 8! would equal 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, over 2!, which is 2 x 1 x 6 x 5 x 4 x 3 x 2 x 1. By multiplying 8! on the top, it equals 40,320, and 2! x 6! on the bottom equals 1,440. Finally, we would divide 40,320 by 1,440, which would equal 28. The Wildcats coach now knows that there are 28 games that will be played in their district this season.
Example
With such a tough season ahead of them, the Wildcats coach knows that his team must have a lot of practice. He decided that the team would play three-on-three games to work on their skills. There are 12 players on the team, and three of them will be chosen for each team. The coach now needs to know how many combinations of teams he could create. To use the equations, the variable n would equal 12 and the r variable would equal 3.
The coach needs to use the equation nCr = n!/r!(n-r)!. The coach then will need to substitute 12 in for n and 3 in for r. Next, he will need to subtract 12-3 = 9. So, he now has 12!/(3! x 9!).
Now, let's expand the factorials. 12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 and 9! x 3! = 3 x 2 x 1 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.
The easiest way to calculate the combination is to cancel out common terms. Since 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 appears on both the top and bottom of the equation, those terms can be canceled out. The coach now has 12 x 11 x 10 on top and 3 x 2 x 1 left on bottom. The coach now multiplies on top, which equals 1,320, and the bottom, which equals 6. Next, the coach divided 1,320 / 6 = 220. The coach could choose 220 different combinations of three-player teams.
Lesson Summary
Just to review, a combination is an arrangement of objects where order does not matter. To calculate a combination, you must use a factorial. A factorial is the product of all the positive integers equal to and less than your number. The equation to calculate the combination is nCr = n!/r!(n-r)!, where n represents the number of items and r represents the number of items being chosen at a time.
When solving the equation, remember that the easiest way to solve is by canceling out common terms on both the top and the bottom of your equation.
Learning Outcomes
Following this lesson, you'll be able to:
- Define combination and factorial
- Write the equation to calculate a combination
- Use the combination equation to solve problems
- Identify a trick to make solving combination problems easier