Question

a hotel reservation code consists of 3 single - digit numbers followed by 3 capital letters.
how many different reservation codes are possible?
answer
6
formula 0.5 points
there are 13 students in a class and 4 of them will be chosen to go on a field trip.
how many ways can these students be chosen?
answer
7
formula 0.5 points
a youth soccer coach much choose 4 of his 9 players to go into a game.

Explanation:

Step1: Calculate the number of ways for the hotel reservation code

For the 3 single - digit numbers: Each digit can be chosen in 10 ways (0 - 9). So the number of ways for the 3 - digit part is \(10\times10\times10=10^{3}\).
For the 3 capital letters: Each letter can be chosen in 26 ways (A - Z). So the number of ways for the 3 - letter part is \(26\times26\times26 = 26^{3}\).
By the multiplication principle, the total number of reservation codes is \(10^{3}\times26^{3}\).

$$
LATEXBLOCK0
$$

Step2: Calculate the number of ways to choose 4 students out of 13

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 13\) and \(r=4\).

$$
LATEXBLOCK1
$$

Step3: Calculate the number of ways for the soccer coach to choose 4 players out of 9

Using the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), with \(n = 9\) and \(r = 4\)

$$
LATEXBLOCK2
$$

Answer:

The number of different hotel reservation codes is \(17576000\). The number of ways to choose 4 students out of 13 is \(715\). The number of ways for the soccer coach to choose 4 players out of 9 is \(126\).