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### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# One hundred tickets, numbered $$1,2,3, \ldots ,100$$, are sold to $$100$$ different people for a drawing. Four different prizes are awarded, including a grand prize (a trip to Tahiti). How many ways are there to award the prizes ifa) there are no restrictions?b) the person holding ticket $$47$$ wins the grand prize?c) the person holding ticket $$47$$ wins one of the prizes?d) the person holding ticket $$47$$ does not win a prize?e) the people holding tickets $$19$$ and $$47$$ both win prizes?f) the people holding tickets $$19\;,\;47$$and $$73$$ all win prizes?g) the people holding tickets $$19\;,\;47\;,\;73$$ and $$97$$ all win prizes?h) none of the people holding tickets $$19\;,\;47\;,\;73$$ and $$97$$ wins a prize?i) the grand prize winner is a person holding ticket $$19\;,\;47\;,\;73$$ or $$97$$?j) the people holding tickets 19 and 47 win prizes, but the people holding tickets $$73$$ and $$97$$ do not win prizes?

a) The number of ways is $$94,109,400$$.

b) The number of ways is $$941,094$$.

c) The number of ways is $$3,764,376$$.

d) The number of ways is $$90,345,024$$.

e) The number of ways is $$114,072$$.

f) The number of ways is $$2328$$.

g) The number of ways is $$24$$.

h) The number of ways is $$79,727,040$$.

i) The number of ways is $$3,764,376$$.

j) The number of ways is $$109,440$$.

See the step by step solution

## Step 1: Given data

Number of tickets $$= 100$$ and number of prizes $$= 4$$.

## Step 2: Concept of Permutation

The word "permutation" refers to the act or process of changing the linear order of an ordered set.

Formula:

$$_n{P_r} = \frac{{n!}}{{(n - r)!}}$$

## Step 3: Calculation for the number of ways to award the prizes if no restrictions

a)

Find the number of ways to select $$4$$ people if there is no restriction.

$$\begin{array}{l}P(100,4) = \frac{{100!}}{{(100 - 4)!}}\\P(100,4) = \frac{{100!}}{{96!}}\\P(100,4) = 100 \times 99 \times 98 \times 97\\P(100,4) = 94,109,400\end{array}$$

Hence, the number of ways is $$94,109,400$$.

## Step 4: Calculation of the number of ways for the person holding ticket $$47$$ wins the grand prize

b)

Here, the ticket number $$47$$ wins the grand prize and there are three prizes to be awarded.

So, $$n = 99\;,\;r = 3$$.

Find the number of ways to select $$3$$ people.

$$\begin{array}{l}P(99,3) = \frac{{99!}}{{(99 - 3)!}}\\P(99,3) = \frac{{99!}}{{96!}}\\P(99,3) = 99 \times 98 \times 97\\P(99,3) = 941,094\end{array}$$

Hence, the number of ways is $$941,094$$.

## Step 5: Calculation of the number of ways for the person holding ticket $$47$$ wins one of the grand prizes

c)

Here, the ticket number $$47$$ wins the one of the prizes to be awarded.

So, the number of ways to ticket 47 wins the prizes $$= \left( {\begin{array}{*{20}{l}}4\\1\end{array}} \right)$$

For the remaining 3 prizes, $$n = 99\;,\;r = 3$$.

Find the number of ways to select $$3$$ people.

$$\begin{array}{l}P(99,3) = \frac{{99!}}{{(99 - 3)!}}\\P(99,3) = \frac{{99!}}{{96!}}\\P(99,3) = 99 \times 98 \times 97\\P(99,3) = 941,094\end{array}$$

Find the total number of ways.

Total number of ways $$= 4 \times 941094$$

Total number of ways $$= 3,764,376$$

Hence, the number of ways is $$3,764,376$$.

## Step 6: Calculation of the number of ways for the person holding ticket $$47$$ does not win a grand

d)

Here, the ticket number $$47$$ does not win the prize. So, there are four prizes to be awarded.

Here, $$n = 99\;,\;r = 4$$.

Find the number of ways to select $$4$$ people.

$$\begin{array}{l}P(99,4) = \frac{{99!}}{{(99 - 4)!}}\\P(99,4) = \frac{{99!}}{{95!}}\\P(99,4) = 99 \times 98 \times 97 \times 95\\P(99,4) = 90,345,024\end{array}$$

Hence, the number of ways is $$90,345,024$$.

## Step 7: Calculation of the number of ways for the people holding tickets $$19$$ and $$47$$ both win prizes

e)

The ticket number $$47$$ wins the one of the prizes.

So, the number of ways to ticket $$47$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}4\\1\end{array}} \right)$$

The ticket number $$19$$ wins the one of the prizes.

So, the number of ways to ticket $$19$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}3\\1\end{array}} \right)$$

For the remaining $$2$$ prizes, $$n = 98\;,\;r = 2$$.

Find the number of ways to select $$2$$ people.

$$\begin{array}{l}P(98,2) = \frac{{98!}}{{(98 - 2)!}}\\P(98,2) = \frac{{98!}}{{96!}}\\P(98,2) = 98 \times 97\\P(98,2) = 9506\end{array}$$

Find the total number of ways.

Total number of ways $$= 4 \times 3 \times 9506$$

Total number of ways $$= 114,072$$

Hence, the number of ways is $$114,072$$.

## Step 8: Calculation of the number of ways for the people holding tickets $$19\;,\;47$$and $$73$$ all win prizes

f)

The ticket number $$47$$ wins the one of the prizes.

So, the number of ways to ticket $$47$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}4\\1\end{array}} \right)$$

The ticket number $$19$$ wins the one of the prizes.

So, the number of ways to ticket $$19$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}3\\1\end{array}} \right)$$

The ticket number $$73$$ wins the one of the prizes.

So, the number of ways to ticket $$73$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}2\\1\end{array}} \right)$$

For the remaining $$1$$ prize, $$n = 97\;,\;r = 1$$.

Find the number of ways to select $$1$$ people.

$$\begin{array}{l}P(97,1) = \frac{{97!}}{{(97 - 1)!}}\\P(97,1) = \frac{{97!}}{{96!}}\\P(97,1) = 97\end{array}$$

Find the total number of ways.

Total number of ways $$= 4 \times 3 \times 2 \times 97$$

Total number of ways $$= 2328$$

Hence, the number of ways is $$2328$$.

## Step 9: Calculation of the number of ways for the people holding tickets $$19\;,\;47\;,\;73$$ and $$97$$ all win prizes

g)

The ticket number $$47$$ wins the one of the prizes.

So, the number of ways to ticket $$47$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}4\\1\end{array}} \right)$$

The ticket number $$19$$ wins the one of the prizes.

So, the number of ways to ticket $$19$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}3\\1\end{array}} \right)$$

The ticket number $$73$$ wins the one of the prizes.

So, the number of ways to ticket $$73$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}2\\1\end{array}} \right)$$

The ticket number $$97$$ wins the one of the prizes.

So, the number of ways to ticket $$97$$ wins the prizes $$= \left( {\begin{array}{*{20}{l}}1\\1\end{array}} \right)$$

Find the total number of ways.

Total number of ways $$= 4 \times 3 \times 2 \times 1$$

Total number of ways $$= 24$$

Hence, the number of ways is $$24$$.

## Step 10: Calculation of the number of ways for none of the people holding tickets $$19\;,\;47\;,\;73$$ and $$97$$ wins a prize

h)

None of the people holding ticket $$19\;,\;47\;,\;73$$and $$97$$ wins a prize.

Since, $$4$$ people do not win the prize.

For the prize, $$n = 96\;,\;r = 4$$.

Find the number of ways to select $$4$$ people.

$$\begin{array}{l}P(96,4) = \frac{{96!}}{{(96 - 4)!}}\\P(96,4) = \frac{{96!}}{{92!}}\\P(96,4) = 96 \times 95 \times 94 \times 93\\P(96,4) = 79,727,040\end{array}$$

Hence, the number of ways is $$\;79,727,040$$.

## Step 11: Calculation of the number of ways for the grand prize winner is a person holding ticket $$19\;,\;47\;,\;73$$ or $$97$$

i)

The number of ways to award a grand prize for $$19\;,\;47\;,\;73$$and $$97$$ is $$= \left( {\begin{array}{*{20}{l}}4\\1\end{array}} \right)$$

For the remaining prizes, $$n = 99\;,\;r = 3$$.

Find the number of ways to select $$1$$ people.

$$\begin{array}{l}P(99,3) = \frac{{99!}}{{(99 - 3)!}}\\P(99,3) = \frac{{99!}}{{96!}}\\P(99,3) = 99 \times 98 \times 97\\P(99,3) = 941,094\end{array}$$

Find the total number of ways.

Total number of ways $$= 4 \times 941094$$

Total number of ways $$= 3,764,376$$

Hence, the number of ways is $$3,764,376$$.

## Step 12: Calculation of the number of ways for the people holding tickets $$19$$ and $$47$$ win prizes but $$73$$ and $$97$$ do not win prizes

j)

The people with ticket $$19$$ and $$47$$ win the prize.

So, $$n = 4\;,\;r = 2$$

Find the number of ways.

$$\begin{array}{l}P(4,2) = \frac{{4!}}{{(4 - 2)!}}\\P(4,2) = \frac{{4!}}{{2!}}\\P(4,2) = 4 \times 3\\P(4,2) = 12\end{array}$$

The people with ticket $$73$$ and $$97$$ do not win the prize.

Now, we have to select $$2$$ people from the remaining $$96$$ people.

So, $$n = 96\;,\;r = 2$$.

Find the number of ways.

$$\begin{array}{l}P(96,2) = \frac{{96!}}{{(96 - 2)!}}\\P(96,2) = \frac{{96!}}{{94!}}\\P(96,2) = 96 \times 95\\P(96,2) = 9120\end{array}$$

Find the total number of ways.

Total number of ways $$= 12 \times 9120$$

Total number of ways $$= 109,440$$

Hence, the number of ways is $$109,440$$.