A container holds a total of 15 spheres, consisting only of emerald and sapphire spheres. If 2 spheres are selected at random without replacement, the probability that both selected spheres have the same color is $\frac{17}{35}$. Determine the total count of sapphire spheres in the container, given that sapphire spheres outnumber emerald spheres.

The correct answer is 9.

Step 1: Define variables and set up the problem

Let $s$ be the total count of sapphire spheres in the container.

Let $e$ be the total count of emerald spheres in the container.

The total number of spheres is given as 15:

$s+e=15$

We are given that sapphire spheres outnumber emerald spheres, which means:

$s>e$

The total number of ways to pick 2 spheres out of 15 without replacement is calculated using combinations:

$\text{Total outcomes}=\frac{15\times 14}{2}=105$

Step 2: Find the number of favorable outcomes for choosing spheres of the same color

Two spheres selected will have the same color if either both are sapphire spheres or both are emerald spheres.

Number of ways to select 2 sapphire spheres:

$\frac{s(s-1)}{2}$

Number of ways to select 2 emerald spheres:

$\frac{e(e-1)}{2}$

The total number of favorable outcomes is the sum of these two cases:

$\text{Favorable outcomes}=\frac{s(s-1)}{2}+\frac{e(e-1)}{2}$

Step 3: Relate favorable outcomes to the given probability

The probability that both selected spheres have the same color is given as $\frac{17}{35}$:

$\frac{\text{Favorable outcomes}}{\text{Total outcomes}}=\frac{17}{35}$

Substitute $\text{Total outcomes}=105$:

$\frac{\text{Favorable outcomes}}{105}=\frac{17}{35}$

Solve for $\text{Favorable outcomes}$:

$\text{Favorable outcomes}=105\times \frac{17}{35}=3\times 17=51$

Step 4: Set up and solve the equation for the number of spheres

Equating the favorable outcomes formula to 51:

$\frac{s(s-1)}{2}+\frac{e(e-1)}{2}=51$

Multiply both sides by 2:

$s(s-1)+e(e-1)=102$

Since $e=15-s$ and $s>e$, we test values of $s$ greater than 7.5:

If $s=9$, then $e=15-9=6$:

$9\times (9-1)+6\times (6-1)=9\times 8+6\times 5=72+30=102$

This matches our equation, and $9>6$ satisfies the condition that sapphire spheres outnumber emerald spheres.

Thus, the total count of sapphire spheres in the container is 9.