7. There are seven clarinet players in the concert band. In how
many ways can they be seated in seven chairs at a concert?
Use the Fundamental Counting Principle.

A. 5,040
C. 840
B. 2,520
D. 210

5,040

Step-by-step explanation:

Fundamental Counting Principle states that if there are m ways of doing a thing and there are n ways of doing other thing  then there are total m*n ways of doing both things.

Example : If there are 5 path to reach a destination and 3 mode of transport ( bike, car, bicycle) . Then are 5 * 3 ways to reach the destination using the different mode of transport available.

Given no of clarinet players = 7

no of chairs = 7

First player can be seated on seven chairs in seven ways

( for illustration let the seat be a , b, c , d , e , f , g. he can sit on any of the chairs)

since one chair is occupied and 6 chair is available

second player can be seated on 6 chairs in 6 ways

for illustration let the seat be a be occupied then b, c , d , e , f , g are  the chairs on which second player can sit.

\

Similarly 3rd, 4th, 5th, 6th and 7th player can be seat on chairs in

5, 4 , 3, 2, and 1 way respectively.

now using  fundamental Counting Principle

since 7 players can sit on chair in

7 , 6, 5 , 4, 3 , 2, 1 ways then together they can be seated in

7 *  6 * 5 *  4 * 3 * 2 * 1 ways = 5, 040 ways

F(x) = x^2 - 3x - 18

Find the values that when multiplied equal -18 and when added equal -3 (6 and -3)
Plug it into f(x) = (x - 1st value)(x - 2nd value)

f(x) = (x - 6)(x + 3)
0 = (x - 6) (x + 3)

(One of the factors has to equal zero, so set them both equal to zero)

x - 6 = 0
x = 6

x + 3 = 0
x = -3

x = 6, x = -3