Combinations
A second useful application of Pascal's triangle is in the calculation of combinations. For example, the number of combinations of n things taken k at a time (called n choose k) can be found by the equation
But this is also the formula for a cell of Pascal's triangle. Rather than performing the calculation, one can simply look up the appropriate entry in the triangle. For example, suppose a basketball team has 10 players and wants to know how many ways there are of selecting 8. Provided we have the first row and the first entry in a row numbered 0, the answer is entry 8 in row 10: 45. That is, the solution of 10 choose 8 is 45.
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Famous quotes containing the word combinations:
“Europe has a set of primary interests, which to us have none, or a very remote relation. Hence she must be engaged in frequent controversies, the causes of which are essentially foreign to our concerns. Hence, therefore, it must be unwise in us to implicate ourselves, by artificial ties, in the ordinary vicissitudes of her politics or the ordinary combinations and collisions of her friendships or enmities.”
—George Washington (1732–1799)
“What is the structure of government that will best guard against the precipitate counsels and factious combinations for unjust purposes, without a sacrifice of the fundamental principle of republicanism?”
—James Madison (1751–1836)
“One way to think about play, is as the process of finding new combinations for known things—combinations that may yield new forms of expression, new inventions, new discoveries, and new solutions....It’s exactly what children’s play seems to be about and explains why so many people have come to think that children’s play is so important a part of childhood—and beyond.”
—Fred Rogers (20th century)