Question 209230: How many different three letter permutations can be formed from the letters in the word clipboard? Answer by Theo(13342) (Show Source):
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permutations are an ordered set. this means that the same elements in a different order constitute a different permutation.
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combinations are an unordered set. this means that the same elements in a different order do not constitute a different combination
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formulas:
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number of permutations is
number of combinations is
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example 1
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form sets of three letter combinations from the letters abcde.
n = 5
x = 3
they are:
abc
abd
abe
acd
ace
ade
bcd
bce
bde
cde
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example 2
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form sets of three letter permutations from the letters abcde.
n = 5
x = 3
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what is happening here is that for each of the combinations we calculated just above, we have 6 permutations. 10 * 6 = 60 permutations.
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you get:
abc * 6
abd * 6
abe * 6
acd * 6
ace * 6
ade * 6
bcd * 6
bce * 6
bde * 6
cde * 6
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if you look at each one of these combinations and make a permutation out of each one then you get the times 6 factor.
take the combination abc for example:
abc
acb
bac
bca
cab
cba
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the one combination of abc yielded 6 permutations of abc.
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with the combination, all those additional permutations are not allowed because they are each composed of the same letters. with the permutation, however, they are allowed because they are in a different order even though they contain the same combination of letters.
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the formula for your problem is number of permutations which is:
the word is clipboard
n = 9
x = 3
n-x = 6
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your formula becomes: = = (9*8*7*6*5*4*3*2*1)/(6*5*4*3*2*1) = 9*8*7 = 504
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the number of combinations would be the number of permutations divided by x! which in this case is 3! which equals 6 making the number of combinations = 504 / 6 = 84.
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one of those combinations is:
cli
this combination leads to 6 permutations which are:
cli
cil
lic
lci
icl
ilc
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that happens for each of the 84 combinations to make the total number of permutations = 504.
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