A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{3!}{2!} \)
\( \frac{3 \times 2 \times 1}{2 \times 1} \)
\( \frac{3}{1} \)
3

If 88% of the town's 43,000 voters cast ballots the number of votes cast is:

(\( \frac{88}{100} \)) x 43,000 = \( \frac{3,784,000}{100} \) = 37,840

The mayor got 52% of the votes cast which is:

(\( \frac{52}{100} \)) x 37,840 = \( \frac{1,967,680}{100} \) = 19,677 votes.

The ratio of football cards to baseball cards is 5:2 and the ratio of baseball cards to basketball cards is 5:1. To solve this problem, we need the baseball card side of each ratio to be equal so we need to rewrite the ratios in terms of a common number of baseball cards. (Think of this like finding the common denominator when adding fractions.) The ratio of football to baseball cards can also be written as 25:10 and the ratio of baseball cards to basketball cards as 10:2. So, the ratio of football cards to basketball cards is football:baseball, baseball:basketball or 25:10, 10:2 which reduces to 25:2.

By buying two shirts, Bob will save $20 x \( \frac{50}{100} \) = \( \frac{$20 x 50}{100} \) = \( \frac{$1000}{100} \) = $10.00 on the second shirt.

So, his total cost will be
$20.00 + ($20.00 - $10.00)
$20.00 + $10.00
$30.00

The number of students taking German or Spanish is 7 + 12 = 19. Of that group of 19, 5 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 19 - 5 = 14 who are taking at least one language. 29 - 14 = 15 students who are not taking either language.