BTGmoderatorDC wrote: ↑

Sat Dec 19, 2020 7:09 pm

A manufacturing company with exactly 20 factories uses a letter coding system to represent each factory. If each factory is represented either by a single letter or by a pair of two distinct letters, what is the least number of letters that can be used to represent all 20 factories. (Assume that the order of the letters in the code does not matter.)

(A) 5

(B) 6

(C) 10

(D) 20

(E) 40


OA B

Solution:

Since the number of factories is 20, the number of letters needed to be used to code them must be less than 20. So let’s try the answer choices, starting with 5.

If we have only 5 letters, the number of single-letter codes that can be formed is 5, and the number of double-letter codes that can be formed is 5C2 = 10 (recall that the letters must be distinct and order doesn’t matter). However, the total number of codes in this case is only 5 + 10 = 15, which is less than the number of factories.

Now, let’s say we have 6 letters. The number of single-letter codes that can be formed is 6, and the number of double-letter codes that can be formed is 6C2 = 15. Therefore, the total number of codes in this case is 6 + 15 = 21, which is (one) more than the number of factories. So 6 letters are sufficient to code all the factories.

Answer: B