Binomial

Binomial simply means “two numbers” and in mathematics it comes up in different ways. And there’s a pattern which also comes up repeatedly in two-number problems. That’s what makes mathematics interesting: it’s the discovery of patterns in what seem to be unrelated problems.

EXAMPLE: How many ways can we select 3 objects out of 7? Here are two numbers 3 and 7.

Let’s work out the answer logically. There are 7 ways to take the first object. That leaves 6, so there are 6 objects left to choose from, leaving 5. Finally, we choose the third object. So the number of ways of choosing 3 objects out of 7 is 7×6×5=210. That means there are 210 permutations. In a permutation the order matters. If the objects were days of the week then Monday, Wednesday and Friday would be a permutation. So would Friday, Monday and Wednesday. But they are the same days in a different order. If we only want the combination of days regardless of order, then we need to find out all the ways we can arrange the same three days. There are 6 ways: MWF, MFW, WMF, WFM, FMW, FWM. 6=1×2×3. So we divide 210 by 6=35 different combinations.

Now take the number 11 which contains two number 1s—a ten and a one. Multiply 11 by itself: 121, then by itself again: 1331, and again 14641. The easy way to multiply by 11 is to write down the first and last digits and add each digit to the one on the right. 121×11=1...1, 1+2=3, 2+1=3, giving 1331. 1331×11=1...1, 1+3=4, 3+3=6, 3+1=4, giving 14641.

14641×11=1...1, 1+4=5, 4+6=10, 6+4=10, 4+1=5, but this would create carryovers so let’s write just the end digits and the sums: 1 5 10 10 5 1. This is row 5 of Pascal’s Triangle.

Here are the first 8 rows, starting at row 0 and ending at row 7:

1

1  1

1  2  1

1  3  3  1

1  4  6  4  1

1  5 10 10  5  1

1  6 15 20 15  6  1

1  7 21 35 35 21  7  1

Add the numbers in each row:

1

2

4

8

16

32

64

128

Note that these are consecutive powers of 2, 2⁷=128. Bi- means 2 and here we have 2 to the power of 7, the number of objects in our list from which we picked three.

Counting from 0 from the left of row 7, 0=1, 1=7, 2=21, 3=35, so 35 is the fourth number on row 7. Remember how many combinations of 3 different objects out of 7 there were? There were 35, and position 3 on row 7 is 35. See the pattern?

The coefficients in the binomial expansion of (x+y)ⁿ copy the nth row of Pascal’s Triangle. We have 35x⁴y³ as the fourth term in the expansion when n=7.

by Top Rated User (695k points)