From the letters of the word SHOWDOWN, how many 4-letter words were formed? (1 Viewer)

[tk8]

Hi!

I got 486 as my answer, but apparently the answer is 606.

Thanks

 

[cossine]

Can you show the working out.

Have you tried considering various cases. e.g. OOXX, WWXX, OOWW, XXXX

edit forget: WWXX

 

[icycledough]

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.