GMAT Practice Question: The expression n! is defined as the product of the integers from 1 through n. If p...
Question
The expression n! is defined as the product of the integers from 1 through n. If p is the product of the integers from 100 through 299 and q is the product of the integers from 200 through 299, which of the following is equal to \frac{p}{q} ?
- 99 !
- 199 !
- \frac{199!}{99!}
- \frac{299!}{99!}
- \frac{299!}{199!}
Topics: factorials, fractions
Solution
Step 1
Express p as a ratio of factorials by observing that multiplying the integers from 100 through 299 is equivalent to dividing 299! by the product of integers from 1 through 99.
p=\frac{299!}{99!}
Step 2
Express q as a ratio of factorials by observing that multiplying the integers from 200 through 299 is equivalent to dividing 299! by the product of integers from 1 through 199.
q=\frac{299!}{199!}
Step 3
Form the quotient \frac{p}{q} using the expressions for p and q obtained in the previous steps.
\frac{p}{q}=\frac{\frac{299!}{99!}}{\frac{299!}{199!}}
Step 4
Convert the complex fraction into a product by multiplying p by the reciprocal of q.
\frac{p}{q}=\frac{299!}{99!}\times\frac{199!}{299!}
Step 5
Cancel the common 299! factors to arrive at the simplified expression.
\frac{p}{q}=\frac{199!}{99!}
Answer
The final answer is \frac{199!}{99!}, which corresponds to choice C.
