[Math] given average score,find the selected numbers

Question

Suppose average score of 46 scores selected from 1,2,3,4,5 is 1.65.Is there any way to find out which are the scores that are selected.

Answer 1

Since $46\cdot 1.65$ is not an integer, apparently the average score $1.65$ was obtained from a precise calculation for $\bar x =\frac1{46}\sum x_i$ by rounding to two decimal places. That is, we may assume that the true value is $1.645\le \bar x< 1.655$. This leads to $75.67\le \sum x_i <76.13$ and since $\sum x_i$ must be an integer, we conclude $\sum x_i=76$.

If we assume that among the 46 numbers there are exactly $n_1$ occurance of 1, $n_2$ occurance of 2 etc., then we find that $$\tag176 = \sum x_i = n_1+2n_2+3n_3+4n_4+5n_5$$ and of course $$\tag246 = n_1+n_2+n_3+n_4+n_5.$$ Subtracting $(1)-(2)$ gives $$\tag3n_2+2n_3+3n_4+4n_5=30.$$ There are mqany possible solutions in nonnegative integers for $(3)$ and they all happen to lead to solutions of the original problem. For example, there might be 30 times 2 and 16 times 1. Or there might be 7 times 5 and once 3 and 38 times 1. Or three each of 2, 3, 4, 5 and 34 times 1. Or, or, or, ...