GMAT Number Properties: If p is the product of the integers…
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 p/q?
- 99!
- 199!
- (199!)/(99!)
- (299!)/(99!)
- (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=(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=(299!)/(199!)
Step 3
Form the quotient p/q using the expressions for p and q obtained in the previous steps.
p/q=((299!)/(99!))/((299!)/(199!))
Step 4
Convert the complex fraction into a product by multiplying p by the reciprocal of q.
p/q=(299!)/(99!)×(199!)/(299!)
Step 5
Cancel the common 299! factors to arrive at the simplified expression.
p/q=(199!)/(99!)
Answer
(199!)/(99!) (C)
