You have 31 out of the total of 350 raffle tickets sold so you have a (\( \frac{31}{350} \)) x 100 = \( \frac{31 \times 100}{350} \) = \( \frac{3100}{350} \) = 9% chance to win the raffle.

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:

\( \frac{6!}{3!} \)
\( \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} \)
\( \frac{6 \times 5 \times 4}{1} \)
\( 6 \times 5 \times 4 \)
120

Christine scored 84% on the test meaning she earned 84% of the possible points on the test. There were 210 possible points on the test so she earned 210 x 0.84 = 177 points. Each question is worth 3 points so she got \( \frac{177}{3} \) = 59 questions right.

By buying two shirts, Alex will save $46 x \( \frac{30}{100} \) = \( \frac{$46 x 30}{100} \) = \( \frac{$1380}{100} \) = $13.80 on the second shirt.

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).