SOLUTION: a product code is made from 3 non-distinct letters from this set { W,Y,T,U,X,R}. how many different codes contain exactly 1 R?

Question 642890: a product code is made from 3 non-distinct letters from this set { W,Y,T,U,X,R}.
how many different codes contain exactly 1 R?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
If repetition is NOT allowed (ex: RTX is allowed, but RTT is NOT allowed)

Case 1) First letter is R

There are 6 letters total. After picking R, you have 5 left. So there are 5*4 = 20 ways to arrange the remaining letters in the remaining two slots.

Case 2) Second letter is R

Same as case 1, but now the R is in the second slot instead of the first. There are still 5 letters left and 5*4 = 20 ways to arrange these 5 remaining letters in the two outer slots.

Case 3) Third letter is R

Same as case 1, but now R is in slot 3. So there are 20 ways to arrange the remaining letters.

So we have 3 cases with 20 ways each giving us 3*20 = 60 ways total.

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OR
If repetition is allowed (ex: RTT is finally allowed), then...

You have 6 letters, but you can only have exactly one R. So once you choose that R (for say the first slot), then you have 5 letters left to choose from for the second slot. Since repetition is allowed, you also have 5 letters left to choose for the 3rd slot.

So you have 5*5 = 25 ways to choose an R for the first slot, then 2 (maybe repeating) letters for the second and third slot.

This can be generalized if R was in any slot. So there are 3 times as many ways, which means that there are 3*25 = 75 different ways

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Summary:

So again, if repetition is NOT allowed, then there are 60 ways to do this.

If repetition is allowed, then there are 75 ways to do this.
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