How many unique sums can be formed by adding any three different numbers from the set $\{4,6,8,10,12,14,16\}\,?$
To find the number of unique sums, we first need to find how many different ways there are to choose three numbers from the set. This is just $\binom{7}{3}=35$. Thus, we just need to find how many different sums we can get when choosing three numbers.
Ordering the numbers in ascending order, we get $4,6,8,10,12,14,16$, and notice that each number must be used so that each sum will be unique. To see this, note that we can get an upper bound by taking the sum of three consecutive numbers, which would be $4+5+6=15$, and our smallest number is greater than this. Thus, the sums must be greater than $15$.
Starting with $4$, we can get $4+5=9$. We can get all other positive integers by adding $4$ to one of the following: $4+5, 4+5+7, 4+5+7+9, 4+5+7+9+11, 4+5+7+9+11+13$.
This gives $9,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27,28$, $29,30,31,32,33,34,35$. These are $\${13}$ distinct sums.
Since the range is $21-9+1=13$, some numbers were found multiple times. For example, $24=4+10+10$ can be found in two ways using $4$, and $11=4+5+2$ can be found in 4 ways using $4$. This gives $\boxed{28}$ as the answer.
To find the number of unique sums that can be formed by adding any three different numbers from the set $\{4,6,8,10,12,14,16\}$, we can use a combination of counting and reasoning.
First, we identify the minimum and maximum possible sums. The smallest sum can be obtained by choosing the three smallest numbers, which are 4, 6, and 8. Thus, the smallest sum is $4 + 6 + 8 = 18$. On the other hand, the largest sum can be obtained by choose the three largest numbers, which are 12, 14, and 16. Therefore, the largest sum is $12 + 14 + 16 = 42$.
Next, we count the number of integers between 18 and 42, inclusive. This can be done by taking the difference between the largest and smallest integers in the range and then adding 1, as we need to include both ends. Hence, there are $42 - 18 + 1 = 25$ integers in the range.
However, not all integers in this range can be obtained by adding three different numbers from the set. This is because the set has only 7 distinct numbers, and every sum we can form must be among those numbers. To find the number of unique sums, we need to find the number of distinct integers among the 25 integers in the range.
By examining the set, we can see that 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, and 42 can be obtained by adding three different numbers from the set. This makes a total of 11 distinct integers.
Therefore, the number of unique sums that can be formed by adding any three different numbers from the set $\{4,6,8,10,12,14,16\}$ is $\boxed{11}$.