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From the letters of the word SHOWDOWN, how many 4-letter words were formed? (1 Viewer)
- Thread starter tk8
- Start date
icycledough
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- Aug 5, 2021
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- 2020
So as mentioned above, it's quite lengthy and you can consider the 4 different cases: XXXX, OOXX, WWXX and WWOO.
With XXXX, there are 6 different letters and 4 spots to choose for. Thus, 6C4 would be 15 different combinations. Each combination has 4P4 variations (for e.g SHOW, SHWO, SWOH, etc). Thus, 4P4 x 15 = 24 x 15 = 360
With OOXX, there are 5 remaining distinct letters to fill out the 2 X spots. Thus, 5C2 = 10 different combinations. Each combination has 4! / 2! (because of O,O repetition); so 5C2 x (4! / 2!) = 10 x 12 = 120
With WWXX, it would be the same as above, so 120.
With OOWW, this would be 4! / (2! x 2!) = 6; this is because of O and W repeating twice.
So, adding them up would get 360 + 120 + 120 + 6 = 606, as needed.
With XXXX, there are 6 different letters and 4 spots to choose for. Thus, 6C4 would be 15 different combinations. Each combination has 4P4 variations (for e.g SHOW, SHWO, SWOH, etc). Thus, 4P4 x 15 = 24 x 15 = 360
With OOXX, there are 5 remaining distinct letters to fill out the 2 X spots. Thus, 5C2 = 10 different combinations. Each combination has 4! / 2! (because of O,O repetition); so 5C2 x (4! / 2!) = 10 x 12 = 120
With WWXX, it would be the same as above, so 120.
With OOWW, this would be 4! / (2! x 2!) = 6; this is because of O and W repeating twice.
So, adding them up would get 360 + 120 + 120 + 6 = 606, as needed.