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Is there a compelling reason to favor factorials over falling powers when teaching permutations and combinations?
Falling powers are intuitive and more general. They also provide formulas for P(n,k) and C(n,k) that are easier to compute and understand!
#MTBoS #maths #mathed
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Falling powers are intuitive and more general. They also provide formulas for P(n,k) and C(n,k) that are easier to compute and understand!
#MTBoS #maths #mathed
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Falling powers are a simple variation on traditional powers:
5^3 = 5*5*5,
5^(3 falling) = 5*4*3 (factors "fall" by one).
In each case, the exponent indicates the number of factors. A falling power is denoted by underlining the exponent k (Knuth's notation). I'll use k_.
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5^3 = 5*5*5,
5^(3 falling) = 5*4*3 (factors "fall" by one).
In each case, the exponent indicates the number of factors. A falling power is denoted by underlining the exponent k (Knuth's notation). I'll use k_.
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Using falling powers, we have
P(5, 3) = 5^3_ instead of 5! / (5-3)! and
C(5, 3) = 5^3_ / 3^3_ instead of 5! / (3!(5-3)!).
Note that computing 5! / (5-3)! leads to 5! / 2! = 5*4*3*2! / 2! = 5*4*3. With falling powers, you skip the simplifying steps. Same for combinations.
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P(5, 3) = 5^3_ instead of 5! / (5-3)! and
C(5, 3) = 5^3_ / 3^3_ instead of 5! / (3!(5-3)!).
Note that computing 5! / (5-3)! leads to 5! / 2! = 5*4*3*2! / 2! = 5*4*3. With falling powers, you skip the simplifying steps. Same for combinations.
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Better still, the falling powers formulas remind us of why they work.
To count permutations of 3 objects taken from 5, we draw a tree with 5 branches, each leading to 4 more branches, each leading to 3 more.
That leads us directly to 5*4*3, not 5*4*3*2*1 / (2*1).
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To count permutations of 3 objects taken from 5, we draw a tree with 5 branches, each leading to 4 more branches, each leading to 3 more.
That leads us directly to 5*4*3, not 5*4*3*2*1 / (2*1).
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To count combinations of 3 objects taken from 5, we count the permutations (5*4*3), then we group together the permutations with the same objects but different orders, so that each such group has 3*2*1 objects.
That leads to (5*4*3) / (3*2*1) not (5*4*3*2*1) / ((3*2*1)*2*1).
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That leads to (5*4*3) / (3*2*1) not (5*4*3*2*1) / ((3*2*1)*2*1).
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To be fair, factorials form an important special case of falling powers (n! = n^n_) and they enjoy a long history, so they're more familiar.
It's not unreasonable to discuss them. But, teaching them exclusively? Please share if you know any good reasons for that!
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It's not unreasonable to discuss them. But, teaching them exclusively? Please share if you know any good reasons for that!
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