A skeet shooting competition awards prizes for each as follows: The first-place winner receives $11$ points; the second-place winner receives $7$ points, the third-place finisher receives $5$ points, and the fourth-place finisher receives $2$ points. No other prizes are awarded. John competes in several rounds of the skeet shooting competition and receives points every time he competes. If the product of all of the points he receives equals $84,700$ , in how many rounds does he participate?
A.$5$
B.$7$
C.$6$
D.$8$

Hint: The question is focusing on the products of all the points achieved by the candidate. One most important thing is mentioned that in every round he got some position, so there is no round without an award point. Since the total product of all the achieved points is $84,700$ . It conveys that $84,700$ is product of all the points that is achieved by the candidate, so try break the products in the form of $11$ , $11$ , $5$ and $2$ because only these points can be obtained by the candidate, so in this way the solution should proceed.

Complete step-by-step answer:
Given: The first-place winner receives $11$ points; the second-place winner receives $7$ points, the third-place finisher receives $5$ points, and the fourth-place finisher receives $2$ points. The product of awards received by the candidate is equal to $84,700$ .
In this question the product of all the awards received $5$is $84,700$ then we try to break it into the multiple of $11$ , $7$ , and $2$ . Now, breaking the $84,700$ in the product form.
$84,700 = 7 \times 11 \times 11 \times 2 \times 2 \times 5 \times 5$
So, 11 points awarded = 2 times
$7$ points awarded = $1$ times
$5$ points awarded = $2$ times
$2$ points awarded = $2$ times
Hence, total number of rounds in which the candidate participate is $(2 + 1 + 2 + 2) = 7$

Note: In this question, students must be careful about a factor of $84,700$ . The product of factors must be in the value of $11,7,5$ and $2$ because only these factors will full-fill our requirement.