Pascal's Triangle

Blaise Pascal was born in France in 1623. He was a child prodigy, who was fascinated by mathematics. When Pascal was 19, he invented the first calculating machine which actually worked. This was something another mathematician named Fibonacci had tried to do before, but failed.

One of the topics which interested Pascal was the likelihood of an event occurring. His interest was triggered by a gambler. The gambler asked Pascal to help him make better guesses, about which scores would be most likely to occur when 2 dice were thrown. In the course of his investigations, Pascal produced a triangular pattern of numbers which now bears his name. The pattern was known to the Chinese 300 years before Pascal, but it was Pascal who fully developed it.

Pascal's triangle is a triangular arrangement of rows. Each row increases by one number. Each row, except the first, begins and ends with a "1" written diagonally. The first row contains only the number 1. Beginning with the second row, each number is the sum of the number written just to the left and right of it above.

The numbers are placed midway between the numbers of the row directly above it. Pascal's triangle is used to show probability and also to figure out combinations.

In the next few sentences I will show probability by explaining coin tossing and other number patterns and puzzles.

Coin Tossing

Examples ...

If you flip 1 coin the possibilities are 1 head (H) or 1 tail (T). This combination of 1 and 1 is the first row in the Pascal Triangle. If you flip two coins you will get a few different results as I will show below...

HH HT
TH
TT
1 2 1

If you flip 3 coins the chances will be as shown...

HHH HHT
HTH
THH
TTH
THT
HTT
TTT
1 3 3 1

This is the pattern in the third row of Pascals Triangle

If you flip four coins the results are...

HHHH HHHT
HHTH
HTHH
THHH
HHTT
TTHH
HTHT
THTH
THHT
HTTH
TTTH
TTHT
THTT
HTTT
TTTT
1 4 6 4 1

As in the fourth row of the triangle are the numbers 1 4 6 4 1

If you flip 5 coins, the results are listed below:

HHHHH HHHHT
HHHTH
HHTHH
HTHHH
THHHH
HHHTT
HHTTH
HTTHH
TTHHH
THTHH
THHTH
THHHT
HTHHT
HHTHT
HTHTH
TTTHH
TTHHT
THHTT
HHTTT
HTHTT
HTTHT
HTTTH
THTTH
TTHTH
THTHT
TTTTH
TTTHT
TTHTT
THTTT
HTTTT
TTTTT
1 5 10 10 5 1

The combination of numbers above of 1 5 10 10 5 1 are the 5th line of the Pascal Triangle. To find the probability of 3 heads and 2 tails:

10 - possible combinations of 3 heads and 2 tails

32
- total possible combinations

(This sum of 32 is found by adding 1 5 10 5 1 which are the total possible outcomes of tossing 5 coins)

Other Number Patterns and Puzzles

Another thing I found out is that if you multiply 11 x 11 you will get 121 which is the 2nd line in Pascal's Triangle.
If you multiply 121 x 11 you get 1331 which is the 3rd line in the triangle.
If you then multiply 1331 x 11 you get 14641 which is the 4th line in Pascal's Triangle, but if you then multiply 14641 x 11 you do not get the 5th line numbers. You get 161051. But after the 5th line it doesn't work anymore.

Another example of probability: Say there are four children Annie, Bob, Carlos, and Danny (A, B, C, D). The teacher wants to choose two of them to hand out books; in how many ways can she choose a pair???

  1. A & B
  2. A & C
  3. A & D
  4. B & C
  5. B & D
  6. C & D
There are six ways to make a choice of a pair.

If the teacher wants to send three students:

  1. A, B, C
  2. A, B, D
  3. A, C, D
  4. B, C, D
If the teacher wants to send a group of "K" children where "K" may range from 0-4; in how many ways will she choose the children

K=0 1 way (There is only one way to send no children)
K=1 4 ways ( A; B; C; D)
K=2 6 ways (like above with Annie, Bob, Carlos, Danny)
K=3 4 ways (above with triplets)
K=4 1 way (there is only one way to send a group of four)

The above numbers (1 4 6 4 1) are the fourth row of numbers in Pascal Triangle.


Bibliography

  1. Lets Investigate Number Patterns by Marion Smoothey
  2. The Joy of Mathematics By Theoni Pappas