Question

1. while organizing a film festival, you must decide which of the 12 movies will be shown on the big screen. you only have the budget to show 5 movies on this screen, and you want to be able to tell moviegoers which order the films will be shown. in how many different ways can you show 5 of the 12 movies on the big screen?

a. 47,520
b. 792
c. 60
d. 95,040

1. a local pizzeria offers 11 toppings for their pizzas and you can choose any 5 of them for one fixed price. how many different pizzas can you order with 5 toppings?

a. 462
b. 120
c. 55,440
d. 332,640

1. at a high school, students volunteer for a committee. how many different 5 - person committees can be chosen?

a. 5,040
b. 120
c. 21
d. 2,520

1. suppose ruth ann has 3 routes she can choose from to get from school to the library, and 5 routes from the library to her home. how many routes are there from ruth ann’s school to her home with a stop at the library?

a. 9
b. 60
c. 15
d. 25

Explanation:

Question 13

Step1: Identify the problem type

This is a permutation problem because the order of the movies matters (we want to tell moviegoers the order). The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 12 \) (total movies) and \( r=5 \) (movies to show).

Step2: Calculate the permutation

First, calculate \( n!=12! = 12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1 \) and \( (n - r)!=(12 - 5)!=7! = 7\times6\times5\times4\times3\times2\times1 \). Then \( P(12,5)=\frac{12!}{7!}=\frac{12\times11\times10\times9\times8\times7!}{7!}=12\times11\times10\times9\times8 \).

Calculate \( 12\times11 = 132 \), \( 132\times10=1320 \), \( 1320\times9 = 11880 \), \( 11880\times8=95040 \)? Wait, no, wait: Wait, \( 12\times11\times10\times9\times8=12\times11 = 132; 132\times10 = 1320; 1320\times9=11880; 11880\times8 = 95040 \)? But wait, the option A is 47520, B is 792, C is 60, D is 95040. Wait, maybe I made a mistake. Wait, no, permutation of 12 taken 5 at a time: \( P(12,5)=12\times11\times10\times9\times8 = 95040 \)? But let's check again. Wait, 12P5: formula is \( \frac{12!}{(12 - 5)!}=\frac{12!}{7!}=12\times11\times10\times9\times8=95040 \). So the answer should be D? Wait, but let me check the options. Option D is 95,040. So that's correct.

Step1: Identify the problem type

This is a combination problem because the order of choosing toppings does not matter. The formula for combinations is \( C(n, r)=\frac{n!}{r!(n - r)!} \), where \( n = 11 \) (total toppings) and \( r = 5 \) (toppings to choose).

Step2: Calculate the combination

First, calculate \( n!=11! = 11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1 \), \( r!=5! = 5\times4\times3\times2\times1 \), and \( (n - r)!=(11 - 5)!=6! = 6\times5\times4\times3\times2\times1 \). Then \( C(11,5)=\frac{11!}{5!×6!}=\frac{11\times10\times9\times8\times7\times6!}{5!×6!}=\frac{11\times10\times9\times8\times7}{5\times4\times3\times2\times1} \).

Calculate numerator: \( 11\times10 = 110; 110\times9 = 990; 990\times8 = 7920; 7920\times7 = 55440 \). Denominator: \( 5\times4 = 20; 20\times3 = 60; 60\times2 = 120; 120\times1 = 120 \). Then \( \frac{55440}{120}=462 \). So the answer is A. 462.

Step1: Identify the problem type

This is a combination problem (order of committee members doesn't matter). The formula is \( C(n, r)=\frac{n!}{r!(n - r)!} \).

Step2: Match with options

Looking at the options, if \( n = 7 \) and \( r = 5 \), then \( C(7,5)=\frac{7!}{5!×(7 - 5)!}=\frac{7!}{5!×2!}=\frac{7\times6}{2\times1}=21 \), which matches option C.

Answer:

D. 95,040

Question 14 (assuming the number of toppings is 11, since it's partially visible "offers 11 toppings")