Today, we will look at a famous number pattern, the Pascal's number triangle.
 |
| Pascal's triangle |
- In the first row, we write the number 1.
- In the next row, we put down two 1s.
- In subsequent rows,
- the first number and the last number are always equal to 1;
- for other number, it will be equal to the sum of two numbers sitting just above it in the previous row.
For example, $1 + 1 = 2$, $1 + 2 = 3$, $2 + 1 = 3$, $1 + 3 = 4$, $3 + 3 = 6$, $3 + 1 = 4$, etc...
 |
| constructing Pascal's triangle: $4 + 6 = 10$ |
We use Pascal's triangle to determine the coefficients in the expansions of $(x+y)^n$ and $(x-y)^n$.
 |
| expand $(x+y)^n$ using Pascal's coefficients |
 |
| expand $(x-y)^n$ using Pascal's coefficients |
We will give each row of the Pascal's triangle an index. The first row of the Pascal's triangle is row 0, the next row is row 1, then row 2, ect..., The numbers on each row will also be given a similar index. The first number is number 0, then number 1, number 2, ect...
If we use $p_{n,k}$ to denote the number of index $k$ on the row $n$ then the Pascal's triangle is constructed based on the following rule $$p_{n-1,k-1} + p_{n-1,k} = p_{n,k}.$$
 |
| the rule to construct Pascal's triangle: $p_{n-1,k-1} + p_{n-1,k} = p_{n,k}$ |
The general formula for $p_{n,k}$ is
$$p_{n,k} = {n \choose k} = \frac{n!}{k! (n-k)!}$$
For example,
$$p_{5,2} = {5 \choose 2} = \frac{5!}{2! 3!} = \frac{1 \times 2 \times 3 \times 4 \times 5}{1 \times 2 \times 1 \times 2 \times 3} = 10.$$
Finally, let us remark that we would normally start the indexing with index 1, then 2, 3, ..., but with Pascal's triangle, we use a special indexing that starts with 0. To help us remember this special indexing, let us read the following story about Sierpinski.
Waclaw Sierpinski was a great Polish mathematician. But one story told us that he was quite an absent-minded man. On one occasion, he and his wife had to move into a new residence. His wife didn't trust him very much, so when they stood down on the street with all their things, she said "Now, you stand here and watch our ten trunks, while I go and get a taxi." She left and left him there, gazing off into space and humming absently. Some minutes later she returned, presumably having called for a taxi. Said Sierpinski (possibly with a glint in his eye): "I thought you said there were ten trunks, but I've only counted to nine." Panicked that some of their possessions might have been stolen from under their noses, his wife demanded "No, they were TEN!", "No, no, no! Just count them. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 !"
Sierpinski had written a very good book on number theory, which was translated to English with the title "250 problems in elementary number theory". If you are interested in number theory, it is a good book to read.
See you again in the next post.
Homework.
1. Explain why the sum of all numbers on the row $n$ of Pascal's triangle is equal to $2^n$.
2. Prove that the number of index $k$ on the row $n$ is equal to $$p_{n,k} = {n \choose k} = \frac{n!}{k! (n-k)!}.$$
3. Prove the identity
$$(x+y)^n = x^n + {n \choose 1} x^{n-1} y + {n \choose 2} x^{n-2} y^2 + \dots + {n \choose {n-2}} x^{2} y^{n-2} + {n \choose 1} x y^{n-1} + y^n$$
$$(x+y)^n = x^n + {n \choose 1} x^{n-1} y + {n \choose 2} x^{n-2} y^2 + \dots + {n \choose {n-2}} x^{2} y^{n-2} + {n \choose 1} x y^{n-1} + y^n$$