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Pascal's Triangle can be used to find combinations.

Suppose you have three socks and want to figure out how many different ways you can choose two of them to wear. You don't care which feet you put them on, it only matters which two socks you pick, so this problem amounts to the question "how many different ways can you choose two objects from a set of three objects?"

The answer can be found in Pascal's Triangle at place 2 in row 3: three ways. [Remember that the first number (1) running down the lefthand side of Pascal's triangle, is always in place 0.]

 

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

  Row 0
 

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

  Row 1
 

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

  Row 2
 

 

 

 

 

  1

 

 3

 

 3

 

 1

 

 

 

 

 

  Row 3
 

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

  Row 4
 

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

  Row 5
 

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

  Row 6
 

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

  Row 7
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

  Row 8
What if you have five socks - how many different ways can you choose two objects from a set of five objects? Find place 2 in row 5: 10 ways.

Because of this choosing property, the binomial coefficient [5:2] is usually read "five choose two."

The probability of choosing one particular combination of two socks is 1/10. For more about permutations and combinations, see the Dr. Math FAQ.

In about 1654 Blaise Pascal began to investigate the chances of getting different values for rolls of the dice, and his discussions with Pierre de Fermat are considered to have laid the foundation for the theory of probability.

The excitement that a gambler feels when making a bet is equal to the amount he might win times the probability of winning it. - Pascal

References

  1. Ask Dr. Math Archives: One Person of Seven Born on Monday,
    Multiple Choice Tests
  2. Ask Dr. Math FAQ: Pascal's Triangle
  3. Ask Dr. Math FAQ: Permutations and Combinations
  4. Pascal's Probability: An Illustrative Problem


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Math Forum * * * * 2 May 1998