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Q7SE

Expert-verifiedFound in: Page 440

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**An ice cream parlour has \({\rm{28}}\) different flavours, \({\rm{8}}\) different kinds of sauce, and \({\rm{12}}\) toppings. **

**a) In how many different ways can a dish of three scoops of ice cream be made where each flavour can be used more than once and the order of the scoops does not matter? **

**b) How many different kinds of small sundaes are there if a small sundae contains one scoop of ice cream, a sauce, and a topping? **

**c) How many different kinds of large sundaes are there if a large sundae contains three scoops of ice cream, where each flavour can be used more than once and the order of the scoops does not matter; two kinds of sauce, where each sauce can be used only once and the order of the sauces does not matter; and three toppings, where each topping can be used only once and the order of the toppings does not matter?**

(a) The number of different ways a dish of three scoops of ice cream can be made where each flavour can be used more than once and the order of the scoops does not matter is \({\rm{4060}}\).

(b) The number of different kinds of small sundaes there are if a small sundae contains one scoop of ice cream, a sauce, and a topping is \({\rm{2688}}\).

(c) The number of different kinds of large sundaes there are if a large sundae contains three scoops of ice cream is \({\rm{25,009,600}}\).

**Product rule: If one event can occur in **\({\rm{m}}\)** ways and a second event can occur in **\({\rm{n}}\)** ways, then the number of ways that the two events can occur in sequence is then **\({\rm{m}} \cdot {\rm{n}}\)**.**

**Definition of permutation (order is important) is –**

**No repetition allowed: **\({\rm{P(n,r) = }}\frac{{{\rm{n!}}}}{{{\rm{(n - r)!}}}}\)

**Repetition allowed: **\({{\rm{n}}^{\rm{r}}}\)

**Definition of combination (order is important) is –**

**No repetition allowed: **\({\rm{C(n,r) = }}\frac{{{\rm{n!}}}}{{{\rm{r!(n - r)!}}}}\)

**Repetition allowed: **\({\rm{C(n + r - 1,r) = }}\frac{{{\rm{(n + r - 1)!}}}}{{{\rm{r!(n - 1)!}}}}\)

**With **\({\rm{n! = n}} \cdot {\rm{(n - 1)}} \cdot ... \cdot {\rm{2}} \cdot {\rm{1}}\)**.**

(a)

The order of the scoops does not matter; thus, it is needed to use the definition of combination.

The scoops are then \({\rm{3}}\) selections from the \({\rm{28}}\) different flavours.

Here, it can be seen \({\rm{n = 28, r = 3}}\).

Repetition is allowed (since each flavour can be used more than once), so substitute the value and calculate –

\(\begin{array}{c}{\rm{C(n + r - 1,r) = C(28 + 3 - 1,103)}}\\{\rm{ = C(30,3)}}\\{\rm{ = }}\frac{{{\rm{30!}}}}{{{\rm{3!(30 - 3)!}}}}\\{\rm{ = }}\frac{{{\rm{30!}}}}{{{\rm{3!27!}}}}\\{\rm{ = 4060}}\end{array}\)

Therefore, the result is obtained as \({\rm{4060}}\).

(b)

It is needed to choose one scoop, one sauce and one topping –

There are –

Scoops: \({\rm{28}}\) ways (flavours)

Sauce: \({\rm{8}}\) ways

Toppings: \({\rm{12}}\) ways

Using the product rule –

\({\rm{28}} \cdot 8 \cdot {\rm{12 = 2688}}\)

Therefore, the result is obtained as \({\rm{2688}}\).

(c)

First consider the calculation for scoops.

The order of the scoops does not matter; thus, it is needed to use the definition of combination.

The scoops are then \({\rm{3}}\) selections from the \({\rm{28}}\) different flavours.

Here, it can be seen \({\rm{n = 28, r = 3}}\).

Repetition is allowed (since each flavour can be used more than once), so substitute the value and calculate –

\(\begin{array}{c}{\rm{C(n + r - 1,r) = C(28 + 3 - 1,103)}}\\{\rm{ = C(30,3)}}\\{\rm{ = }}\frac{{{\rm{30!}}}}{{{\rm{3!(30 - 3)!}}}}\\{\rm{ = }}\frac{{{\rm{30!}}}}{{{\rm{3!27!}}}}\\{\rm{ = 4060}}\end{array}\)

Now consider the calculation for sauces.

The order of the sauces does not matter; thus, it is needed to use the definition of combination.

The scoops are then \({\rm{2}}\) selections from the \({\rm{8}}\) different flavours.

Here, it can be seen \({\rm{n = 8, r = 2}}\).

Repetition is not allowed (since each sauce can be used only once), so substitute the value and calculate –

\(\begin{array}{c}{\rm{C(8,2) = }}\frac{{{\rm{8!}}}}{{{\rm{2!(8 - 2)!}}}}\\{\rm{ = }}\frac{{{\rm{8!}}}}{{{\rm{2!6!}}}}\\{\rm{ = 28}}\end{array}\)

Now consider the calculation for toppings.

The order of the toppings does not matter; thus, it is needed to use the definition of combination.

The scoops are then \({\rm{3}}\) selections from the \({\rm{12}}\) different flavours.

Here, it can be seen \({\rm{n = 12, r = 3}}\).

Repetition is not allowed (since each topping can be used only once), so substitute the value and calculate –

\(\begin{array}{c}{\rm{C(12,3) = }}\frac{{{\rm{12!}}}}{{{\rm{3!(12 - 3)!}}}}\\{\rm{ = }}\frac{{{\rm{12!}}}}{{{\rm{3!9!}}}}\\{\rm{ = 220}}\end{array}\)

Using the product rule –

\({\rm{4060}} \cdot 28 \cdot {\rm{220 = 25,009,600}}\)

Therefore, the result is obtained as \({\rm{25,009,600}}\).

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