Question

3 fill in the blank 15 points (a) a pizza shop offers twelve different toppings. how many different three - topping pizzas can be formed with the twelve toppings? type your answer... (b) your english teacher has asked you to select 3 novels from a list of 10 to read as an independent project. in how many ways can you choose which books to read? type your answer... a restaurant serves omelets that can be ordered with any of the ingredients shown: omelets $4 (plus $0.50 for each ingredient) vegetarian toppings: green pepper, red pepper, onion, mushroom, tomato, cheese meat toppings: ham, bacon, sausage, steak (c) suppose you want exactly 2 vegetarian ingredients and 1 meat ingredient in your omelet. how many different types of omelets can you order? type your answer... (d) suppose you can afford at most 3 ingredients in your omelet. how many different types of omelets can you order? type your answer...

Explanation:

Part (a)

We need to find the number of ways to choose 3 toppings out of 12. This is a combination problem since the order of the toppings does not matter. The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 12 \) and \( k=3 \).

Step 1: Identify the formula and values

\( n = 12 \), \( k = 3 \), so we use \( C(12,3)=\frac{12!}{3!(12 - 3)!} \)

Step 2: Simplify the factorials

\( 12! = 12\times11\times10\times9! \), \( 3! = 3\times2\times1 \), \( 9! \) cancels out. So \( C(12,3)=\frac{12\times11\times10}{3\times2\times1} \)

Step 3: Calculate the value

\( \frac{12\times11\times10}{6}=\frac{1320}{6} = 220 \)

Part (b)

We need to choose 3 novels out of 10. Again, this is a combination problem with \( n = 10 \) and \( k = 3 \)

Step 1: Identify the formula and values

\( C(10,3)=\frac{10!}{3!(10 - 3)!} \)

Step 2: Simplify the factorials

\( 10! = 10\times9\times8\times7! \), \( 3! = 3\times2\times1 \), \( 7! \) cancels out. So \( C(10,3)=\frac{10\times9\times8}{3\times2\times1} \)

Step 3: Calculate the value

\( \frac{720}{6}=120 \)

Part (c)

We need to choose 2 vegetarian ingredients out of 6 (green pepper, red pepper, onion, mushroom, tomato, cheese) and 1 meat ingredient out of 4 (ham, bacon, sausage, steak). We use the multiplication principle of counting. First, find the number of ways to choose vegetarian ingredients \( C(6,2) \) and the number of ways to choose meat ingredients \( C(4,1) \), then multiply them.

Step 1: Calculate \( C(6,2) \)

\( C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5\times4!}{2\times1\times4!}=\frac{30}{2} = 15 \)

Step 2: Calculate \( C(4,1) \)

\( C(4,1)=\frac{4!}{1!(4 - 1)!}=\frac{4\times3!}{1\times3!}=4 \)

Step 3: Multiply the two results

\( 15\times4 = 60 \)

Part (d)

We can have 0, 1, 2, or 3 ingredients. First, find the total number of ingredients: vegetarian (6) + meat (4) = 10 ingredients.

Step 1: Calculate the number of ways for 0 ingredients

\( C(10,0)=1 \) (by definition, choosing 0 items from n items is 1)

Step 2: Calculate the number of ways for 1 ingredient

\( C(10,1)=\frac{10!}{1!(10 - 1)!}=\frac{10\times9!}{1\times9!}=10 \)

Step 3: Calculate the number of ways for 2 ingredients

\( C(10,2)=\frac{10!}{2!(10 - 2)!}=\frac{10\times9\times8!}{2\times1\times8!}=\frac{90}{2}=45 \)

Step 4: Calculate the number of ways for 3 ingredients

\( C(10,3)=\frac{10!}{3!(10 - 3)!}=\frac{10\times9\times8\times7!}{3\times2\times1\times7!}=\frac{720}{6} = 120 \)

Step 5: Sum up the results

\( 1+10 + 45+120=176 \)

Answer:

(a) \(\boldsymbol{220}\)

(b) \(\boldsymbol{120}\)

(c) \(\boldsymbol{60}\)

(d) \(\boldsymbol{176}\)