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 the variables and understand the given conditions
Let $s$ represent the number of sapphire spheres and $e$ represent the number of emerald spheres in the container.
We are given that the total number of spheres is 15. Therefore:

$s+e=15$

We can express the number of emerald spheres in terms of sapphire spheres as:

$e=15-s$

We are also given that sapphire spheres outnumber emerald spheres, which gives the inequality:

$s>e$

Step 2: Calculate the total number of ways to pick 2 spheres
The total number of ways to select 2 spheres out of 15 without replacement is given by the combination formula:

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

Step 3: Determine the number of favorable outcomes
The two selected spheres have the same color if both chosen spheres are sapphire OR both chosen spheres are emerald.
The number of ways to choose 2 sapphire spheres from $s$ spheres is:

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

The number of ways to choose 2 emerald spheres from $e$ spheres is:

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

Adding these two options together gives the total number of favorable outcomes:

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

Now, substitute $e=15-s$ into the expression:

$\text{Numerator}=s^2-s+{(15-s)}^2-(15-s)$

Expand and simplify the numerator:

$\text{Numerator}=s^2-s+225-30s+s^2-15+s=2s^2-30s+210$

Dividing the numerator by 2 simplifies the favorable outcomes to:

$\text{Favorable Outcomes}=s^2-15s+105$

Step 4: Set up the probability equation
The probability of picking two spheres of the same color is the ratio of favorable outcomes to total outcomes. We are given that this probability equals $\frac{17}{35}$:

$\frac{s^2-15s+105}{105}=\frac{17}{35}$

Multiply both sides of the equation by 105:

$s^2-15s+105=105\times \frac{17}{35}$

$s^2-15s+105=3\times 17$

$s^2-15s+105=51$

Subtract 51 from both sides to form a standard quadratic equation:

$s^2-15s+54=0$

Step 5: Solve for $s$
Factor the quadratic equation:

$(s-9)(s-6)=0$

This yields two possible values for $s$:

$s=9$ or $s=6$

Step 6: Apply the problem constraints
If $s=9$, then $e=15-9=6$. Since $9>6$, this satisfies the condition that sapphire spheres outnumber emerald spheres.
If $s=6$, then $e=15-6=9$. This would mean emerald spheres outnumber sapphire spheres, violating the condition.

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