Answer :
Below is a step‐by‐step explanation and solution for each scenario:
------------------------------------------------------------
1) There are 30 applicants for two different jobs: computer programmer and software tester.
Since the jobs are distinct, the order matters. This is a permutation problem where we choose 2 out of 30. The number of possibilities is
[tex]$$
P(30,2) = 30 \times 29 = 870.
$$[/tex]
------------------------------------------------------------
2) The batting order for seven players on a 10 person team.
Here we want to select 7 players from 10 and arrange them in order. This is given by
[tex]$$
P(10,7) = \frac{10!}{(10-7)!} = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4.
$$[/tex]
Calculating step-by-step:
[tex]\[
10 \times 9 = 90
\][/tex]
[tex]\[
90 \times 8 = 720
\][/tex]
[tex]\[
720 \times 7 = 5040
\][/tex]
[tex]\[
5040 \times 6 = 30240
\][/tex]
[tex]\[
30240 \times 5 = 151200
\][/tex]
[tex]\[
151200 \times 4 = 604800.
\][/tex]
Thus, there are
[tex]$$
604800 \text{ possible batting orders.}
$$[/tex]
------------------------------------------------------------
3) There are 80 politicians at a meeting, and each one gives a Valentine's Day card to everyone else.
Each politician gives a card to [tex]$80 - 1 = 79$[/tex] other politicians, so the total is
[tex]$$
80 \times 79 = 6320 \text{ cards.}
$$[/tex]
------------------------------------------------------------
4) Mofor and Ashley are planning trips to four countries out of 8 countries, with each trip (of different duration) being distinct.
Since the order matters, we calculate the number of ways to choose and order 4 countries out of 8:
[tex]$$
P(8,4) = 8 \times 7 \times 6 \times 5.
$$[/tex]
Now, compute:
[tex]\[
8 \times 7 = 56,
\][/tex]
[tex]\[
56 \times 6 = 336,
\][/tex]
[tex]\[
336 \times 5 = 1680.
\][/tex]
So, there are
[tex]$$
1680 \text{ possible ways.}
$$[/tex]
------------------------------------------------------------
5) A group of 45 people are going to run a race that awards gold, silver, and bronze medals.
This is a permutation of 3 out of 45 as the order (medal ranking) matters:
[tex]$$
P(45,3) = 45 \times 44 \times 43.
$$[/tex]
Calculate step-by-step:
[tex]\[
45 \times 44 = 1980,
\][/tex]
[tex]\[
1980 \times 43 = 85140.
\][/tex]
Thus, there are
[tex]$$
85140 \text{ ways.}
$$[/tex]
------------------------------------------------------------
6) There are 260 politicians at a meeting, and each one gives a card to everyone else.
Each one gives a card to [tex]$260 - 1 = 259$[/tex] others. The total number of cards is
[tex]$$
260 \times 259.
$$[/tex]
Calculating:
[tex]\[
260 \times 259 = 260 \times (260 - 1) = 67600 - 260 = 67340.
\][/tex]
So, there are
[tex]$$
67340 \text{ cards given.}
$$[/tex]
------------------------------------------------------------
7) A team of 13 softball players needs to choose a captain, a co-captain, and a treasurer.
The roles are distinct so we use permutations:
[tex]$$
P(13,3) = 13 \times 12 \times 11.
$$[/tex]
Computing:
[tex]\[
13 \times 12 = 156,
\][/tex]
[tex]\[
156 \times 11 = 1716.
\][/tex]
Thus, there are
[tex]$$
1716 \text{ ways.}
$$[/tex]
------------------------------------------------------------
Summary of Answers:
1) [tex]$\boxed{870}$[/tex]
2) [tex]$\boxed{604800}$[/tex]
3) [tex]$\boxed{6320}$[/tex]
4) [tex]$\boxed{1680}$[/tex]
5) [tex]$\boxed{85140}$[/tex]
6) [tex]$\boxed{67340}$[/tex]
7) [tex]$\boxed{1716}$[/tex]
Each of the above values represents the number of possibilities for the respective scenario.
------------------------------------------------------------
1) There are 30 applicants for two different jobs: computer programmer and software tester.
Since the jobs are distinct, the order matters. This is a permutation problem where we choose 2 out of 30. The number of possibilities is
[tex]$$
P(30,2) = 30 \times 29 = 870.
$$[/tex]
------------------------------------------------------------
2) The batting order for seven players on a 10 person team.
Here we want to select 7 players from 10 and arrange them in order. This is given by
[tex]$$
P(10,7) = \frac{10!}{(10-7)!} = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4.
$$[/tex]
Calculating step-by-step:
[tex]\[
10 \times 9 = 90
\][/tex]
[tex]\[
90 \times 8 = 720
\][/tex]
[tex]\[
720 \times 7 = 5040
\][/tex]
[tex]\[
5040 \times 6 = 30240
\][/tex]
[tex]\[
30240 \times 5 = 151200
\][/tex]
[tex]\[
151200 \times 4 = 604800.
\][/tex]
Thus, there are
[tex]$$
604800 \text{ possible batting orders.}
$$[/tex]
------------------------------------------------------------
3) There are 80 politicians at a meeting, and each one gives a Valentine's Day card to everyone else.
Each politician gives a card to [tex]$80 - 1 = 79$[/tex] other politicians, so the total is
[tex]$$
80 \times 79 = 6320 \text{ cards.}
$$[/tex]
------------------------------------------------------------
4) Mofor and Ashley are planning trips to four countries out of 8 countries, with each trip (of different duration) being distinct.
Since the order matters, we calculate the number of ways to choose and order 4 countries out of 8:
[tex]$$
P(8,4) = 8 \times 7 \times 6 \times 5.
$$[/tex]
Now, compute:
[tex]\[
8 \times 7 = 56,
\][/tex]
[tex]\[
56 \times 6 = 336,
\][/tex]
[tex]\[
336 \times 5 = 1680.
\][/tex]
So, there are
[tex]$$
1680 \text{ possible ways.}
$$[/tex]
------------------------------------------------------------
5) A group of 45 people are going to run a race that awards gold, silver, and bronze medals.
This is a permutation of 3 out of 45 as the order (medal ranking) matters:
[tex]$$
P(45,3) = 45 \times 44 \times 43.
$$[/tex]
Calculate step-by-step:
[tex]\[
45 \times 44 = 1980,
\][/tex]
[tex]\[
1980 \times 43 = 85140.
\][/tex]
Thus, there are
[tex]$$
85140 \text{ ways.}
$$[/tex]
------------------------------------------------------------
6) There are 260 politicians at a meeting, and each one gives a card to everyone else.
Each one gives a card to [tex]$260 - 1 = 259$[/tex] others. The total number of cards is
[tex]$$
260 \times 259.
$$[/tex]
Calculating:
[tex]\[
260 \times 259 = 260 \times (260 - 1) = 67600 - 260 = 67340.
\][/tex]
So, there are
[tex]$$
67340 \text{ cards given.}
$$[/tex]
------------------------------------------------------------
7) A team of 13 softball players needs to choose a captain, a co-captain, and a treasurer.
The roles are distinct so we use permutations:
[tex]$$
P(13,3) = 13 \times 12 \times 11.
$$[/tex]
Computing:
[tex]\[
13 \times 12 = 156,
\][/tex]
[tex]\[
156 \times 11 = 1716.
\][/tex]
Thus, there are
[tex]$$
1716 \text{ ways.}
$$[/tex]
------------------------------------------------------------
Summary of Answers:
1) [tex]$\boxed{870}$[/tex]
2) [tex]$\boxed{604800}$[/tex]
3) [tex]$\boxed{6320}$[/tex]
4) [tex]$\boxed{1680}$[/tex]
5) [tex]$\boxed{85140}$[/tex]
6) [tex]$\boxed{67340}$[/tex]
7) [tex]$\boxed{1716}$[/tex]
Each of the above values represents the number of possibilities for the respective scenario.