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The Relationship Between Pascal's Triangle & Combinations

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Pascal's triangle shows many important mathematical concepts like the counting numbers and the binomial coefficients. In this lesson, we show how Pascal's triangle is related to combinations.

The Relationship Between Pascal's Triangle and Combinations

Ever notice the variety of fruit juices sold at the supermarket? There are all sorts of combinations, like mango-banana-orange and apple-strawberry-orange. Jeremy wonders how many different combinations could be made from five fruits. His plan is to take three at a time. This number of combinations is related to the numbers that appear in Pascal's triangle.

Counting the Number of Combinations

A combination is a grouping of items from a larger collection of those items. In the example of fruit juices, Jeremy's larger collection of 5 fruits could be: apple, orange, banana, strawberry and mango. He plans to take 3 at a time. It does not matter which order the fruits are combined; the juice will taste the same. Also, there will be no repetition of the same fruit for a particular juice combination. Jeremy wants three distinct fruits in each combination. When the order does not matter, we have a combination. Some combinations have repetition and others do not. In Jeremy's case, the ordering of fruits does not matter and there is no repetition. Just for fun, let's list all the possibilities:

1. Apple-Orange-Banana

2. Apple-Orange-Strawberry

3. Apple-Orange-Mango

4. Apple-Banana-Strawberry

5. Apple-Banana-Mango

6. Apple-Strawberry-Mango

7. Orange-Banana-Strawberry

8. Orange-Banana-Mango

9. Orange-Strawberry-Mango

10. Banana-Strawberry-Mango

That gives us 10 fruit juice combinations.

The mathematical formula for the number of combinations without repetition is N!/(n!(N-n)!). In Jeremy's case, N is 5 and n=3. Thus, the number of combinations is 5!/(3!(5-3)!). The exclamation point stands for factorial and 3! = 3(2)(1) = 6. The 5! = 5(4)(3)(2)(1) = 120 and (5-3)! is 2! = 2(1) = 2. Thus, the number of combinations is 5!/((3!)(5-3)!) = 120/((6)(2)) =120/12 = 10.

One short way to write the number of combinations is


the_N_take_n_expression


This is read as '' N take n '' and is evaluated using factorials. For example, if Jeremy decides to include four fruits in each juice combination, this will be 5 take 4:


the_5_take_4_expression


See how 4! Is 4(3)(2)(1) = 24. Also, it's easy to see 5 is the correct number of possible juice combos: Apple-Orange-Banana-Strawberry, Apple-Orange-Banana-Mango, Apple-Orange-Strawberry-Mango, Apple-Banana-Strawberry-Mango and Orange-Banana-Strawberry-Mango.

Pascal's Triangle

To construct Pascal's triangle, start with a 1. Then, in the next row, write a 1 and 1. It's good to have spacing between the numbers. In the third row, we have 1 and 1 on the outside slopes. The 2 comes from adding the two numbers above and adjacent. Thus, we are adding the number on the left, 1, with the number on the right, 1, to get 1 + 1 = 2.

In the next row, the 3 comes from adding the 1 and the 2. This particular Pascal's triangle stopped at 1 5 10 10 5 1, but we could have continued indefinitely.


Pascal


The top row is called ''row 0'' and the first number on the left in each row is the ''0''th number. Remember how 5 take 3 is 10? Count down the Pascal's triangle until you reach row 5 (starting with 0 for the first row). Then, starting from the left, count over to the 3rd place (the first number is at place 0). What number do you see? Right, it's 10, the same result calculated using factorials.

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