How many Anas?

Probability Level 3

Aldo has all the letters in the alphabet in three different sizes: Large, medium and small. Using two different sizes of letters, Aldo wants to write the name of her friend Ana. In how many ways can he write it? Note: The name must have at least one letter of a different size.

20 27 24 18

Juan Alberto Regalado Galván by Juan Alberto Regalado Ga… , 32

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2 solutions

All the possible ways, N = 3 3 = 27 N=3^3=27 .

The number of ways only one size is used, N 1 = 3 N_1=3 .

The numbers of ways three sizes are used, N 3 = 3 ! = 6 N_3=3!=6 .

Therefore, the number of ways two sizes are used, N 2 = N − N 1 − N 3 = 27 − 3 − 6 = 18 N_2=N-N_1-N_3=27-3-6=\boxed{18} .

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The correct solution is 18. Suppose we have only upper case or lower case letters A , a , n and N . We can count how many combinations we can make to form the name Ana . It's 2 3 = 8 2^{3} = 8 because we have 2 types of letters and 3 spaces to put them. We can even write them down:

  • ana
  • anA
  • aNa
  • aNA
  • Ana
  • AnA
  • ANa
  • ANA

But there are two of them which have only 1 type of letter on it, namely ana and ANA , so this 2 doesn't count.

Now, back to the original problem, we can only mix letters from two different sizes, and because there are 3 ways to pick pairs of sizes from 3 different letters sizes; and we have 8 − 2 = 6 8 - 2 = 6 combinations for each pair of sizes, then we have a total of 3 × ( 2 3 − 2 ) = 18 3 \times (2^{3} -2) = 18 possible combinations.

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