How many permutations of three letters out of the word BANANA can be made?

Question

Help!

How many permutations of $3$ letters out of the word BANANA can be made?

The full letter BANANA will only give you $60$ different ways: ${ 6 ! } / ( 1 ! 3 ! 2 ! ) = 720 / 12 = 60 $

But since they only want $3$ letters, the probability should be double

Is the answer $6P_3 = 120$ or $(6!/3!\times2!)/2 = 120$?

Answer 1

Add up the following:

• Number of permutations of AAA: $\frac{3!}{3!}=1$
• Number of permutations of AAB: $\frac{3!}{2!1!}=3$
• Number of permutations of AAN: $\frac{3!}{2!1!}=3$
• Number of permutations of ABN: $\frac{3!}{1!1!1!}=6$
• Number of permutations of ANN: $\frac{3!}{1!2!}=3$
• Number of permutations of BNN: $\frac{3!}{1!2!}=3$