Permutations (1 Viewer)

[goobi]

Question:
Find the number of code words involving 4 distinct letters chosen from the letters in the word forest
(a) if there are no restrictions
(b) if no code word contains both the letters R and S.

(a)Solution: 6P4=360 (I got this one)

However I'm stuck with part (b).

(b)Solution: 360 - 12 x 4P2 = 216

I do know that the read part is the number of permutations with both R and S.
But I do not really get how it is derived...

Thanks for any help!!

 

[gurmies]

Personally, combinations make more intuitive sense to me.

We want to find how many different 4 letter words can be made containing R and S (and subtract it from 360). Since we already know it contains these two letters, we need to choose 2 letters from the remaining 4 (i.e. 4C2). Then we arrange the entire set with multiplication by 4!. So we have 360 - 4C2*4! = 216.

[Note: 4C2*4! = (4!*4!)/(2!*2!) = 4P2*(4!/2!) = 12*4P2]

 

[goobi]

Thank you so much! But I'm interested in how it works with permutations. Can you, or anyone please explain it?

 

[gurmies]

Thank you so much! But I'm interested in how it works with permutations. Can you, or anyone please explain it?

It works in much the same way. Once again, you are interested in two letters out of four - arranging them gives 4P2. Now you need to arrange these two with the two you've already chosen. Treat it as if you haven't arranged anything yet - 4! * 4P2. Since you've actually arranged two already in the 4P2 term, you divide out by 2!. This yields 4P2 * 4!/2! = 4P2*12. As you can see, the intuition here is a little hazy. Use combinations where possible.

 

[goobi]

Thank you so much! I got it now!!!