You have 24 out of the total of 400 raffle tickets sold so you have a (\( \frac{24}{400} \)) x 100 = \( \frac{24 \times 100}{400} \) = \( \frac{2400}{400} \) = 6% chance to win the raffle.

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{8z^7}{7z^3} \)
\( \frac{8}{7} \) z^{(7 - 3)}
1\(\frac{1}{7}\)z⁴

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [12, 24, 36, 48, 60] making 12 the smallest multiple 6 and 12 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{4 x 2}{6 x 2} \) - \( \frac{5 x 1}{12 x 1} \)

\( \frac{8}{12} \) - \( \frac{5}{12} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{8 - 5}{12} \) = \( \frac{3}{12} \) = \(\frac{1}{4}\)

A ratio of 2:1 means that there are 2 home fans for every one visiting fan. So, of every 3 fans, 2 are home fans and \( \frac{2}{3} \) of every fan in the stadium is a home fan:

40,000 fans x \( \frac{2}{3} \) = \( \frac{80000}{3} \) = 26,667 fans.

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:

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