Q: permutations (1 Viewer)

freaking_out

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this permutation stuff is driving me nut anyway my question is the following:

find the number of 5 letter words which can be made from the letters of the word HONGKONG.

(the thing thats troubling me is the repetition of some letters)


thanx in advanced:D
 

spice girl

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Originally posted by freaking_out
this permutation stuff is driving me nut anyway my question is the following:

find the number of 5 letter words which can be made from the letters of the word HONGKONG.

(the thing thats troubling me is the repetition of some letters)


thanx in advanced:D
whoa

ok whenever you make words from some but not all of repeated letters such as this, you've got to case bash.

Here we have 5 different letters: H,O,N,G,K
we have pair O,N,G

Notice that you can have 2 pairs at max (since we have 5 letter words).

So:
case1: no repeated letters: 5! = 120

case2: one pair, 3 other different letters: there are 3 types of pairs you can have, each type (e.g. double O), you can internally arrange into combination 5C2, and you have 4 different letters to choose the remaining 3.

thus case2: 3 * (5C2) * 4 * 3 * 2 = 720

case 3: two pairs + 1 different letter: there are 3 'dbl pairs': OONN, OOGG, NNGG. Internal arrangement for each is (5C2)*(3C2) (choose 2 spots from 5 for first pair, choose 2 spots from 3 for second pair), and there are 3 options for the last letter:

thus case3: 3 * (5C2) * (3C2) * 3 = 270

Thus total = 1110

edit: Go Hong Kong!
 

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