How many integers can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots, 46\}$ ?
(A) 45
(B) 37
(C) 36
(D) 43
(E) 42
How many integers can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots, 46\}$ ?
(A) 45
(B) 37
(C) 36
(D) 43
(E) 42

## Solution

Since each number is of the form $1+3 n, n=1,2,3, \ldots, 15$, the sum of the three numbers will be of the form $3+3 k+3 l+3 m$ where $k, l$ and $m$ are chosen from $\{1,2,3, \ldots, 15\}$. So the question is equivalent to the easier question of, 'How many distinct integers can be formed by adding three numbers from, $\{1,2,3, \ldots, 15\}$ ?'

The smallest is $1+2+3=6$ and the largest is $13+14+15=42$.

It is clearly possible to get every sum between 6 and 42 by:

(a) increasing the sum by one replacing a number with one that is 1 larger or,

(b) decreasing the sum by one by decreasing one of the addends by 1 .

Thus all the integers from 6 to 42 inclusive can be formed.

This is the same as asking, 'How many integers are there between 1 and 37 inclusive?' The answer, of course, is 37 .