Answer:
The number of ways to select 3 cars and 5 trucks is 69,06,900.
Step-by-step explanation:
In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.
The formula to compute the combinations of k items from n is given by the formula:
![{n\choose k}=\frac{n!}{k!(n-k)!}](https://tex.z-dn.net/?f=%7Bn%5Cchoose%20k%7D%3D%5Cfrac%7Bn%21%7D%7Bk%21%28n-k%29%21%7D)
It is provided that:
Number of different cars, <em>n</em> (C) = 25.
Number of different trucks, <em>n</em> (T) = 15.
Devin selects 8 vehicles to display on the shelf in his room.
Compute the number of ways in which he can select 3 cars from 25 different cars as follows:
![{25\choose 3}=\frac{25!}{3!(25-3)!}=\frac{25\times24\times23\times22!}{3!\times22!}=2300](https://tex.z-dn.net/?f=%7B25%5Cchoose%203%7D%3D%5Cfrac%7B25%21%7D%7B3%21%2825-3%29%21%7D%3D%5Cfrac%7B25%5Ctimes24%5Ctimes23%5Ctimes22%21%7D%7B3%21%5Ctimes22%21%7D%3D2300)
There are 2300 ways to select 3 cars.
Compute the number of ways in which he can select 5 trucks from 25 different trucks as follows:
![{15\choose 5}=\frac{15!}{5!(15-5)!}=\frac{15\times14\times13\times12\times 11\times10!}{5!\times10!}=3003](https://tex.z-dn.net/?f=%7B15%5Cchoose%205%7D%3D%5Cfrac%7B15%21%7D%7B5%21%2815-5%29%21%7D%3D%5Cfrac%7B15%5Ctimes14%5Ctimes13%5Ctimes12%5Ctimes%2011%5Ctimes10%21%7D%7B5%21%5Ctimes10%21%7D%3D3003)
There are 3003 ways to select 5 trucks.
Compute the total number of ways to select 3 cars and 5 trucks as follows:
n (3 cars and 5 trucks) = n (3 cars) × n (5 trucks)
![={25\choose 3}\times {15\choose 5}\\=2300\times 3003\\=6906900](https://tex.z-dn.net/?f=%3D%7B25%5Cchoose%203%7D%5Ctimes%20%7B15%5Cchoose%205%7D%5C%5C%3D2300%5Ctimes%203003%5C%5C%3D6906900)
Thus, the total number of ways to select 3 cars and 5 trucks is 69,06,900.