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 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
The final answer is 199!/99!, which corresponds to choice C.
