At a game show, there are 6 people (including you and your friend) in the front row. The host randomly chooses 3 people from the front row to be contestants. The order in which they are chosen does not matter. How many ways can you and your friend both be chosen?
A. ${ }_6 P_3=120$
B. ${ }_4 C_1=4$
C. ${ }_6 C_3=20$
D. ${ }_6 P_2=30
Answer (1)
You and your friend must both be chosen, leaving 1 spot to fill from the remaining 4 people.
Calculate the number of ways to choose 1 person from 4 using combinations: 4 C 1 .
4 C 1 = 1 ! ( 4 − 1 )! 4 ! = 4 .
There are 4 ways for you and your friend to both be chosen.
Explanation
Understand the problem We are given a problem where we need to find the number of ways to choose 3 people out of 6, with the condition that you and your friend must both be chosen. The order in which the contestants are chosen does not matter.
Determine the remaining choices Since you and your friend are already chosen, we need to choose 1 more person from the remaining 4 people. This is because there are 3 spots to fill, and 2 are already taken by you and your friend.
Calculate the combinations We need to calculate the number of ways to choose 1 person from the remaining 4 people. This can be calculated using combinations, denoted as 4 C 1 or ( 1 4 ) . The formula for combinations is given by: n C r = r ! ( n − r )! n ! In our case, n = 4 and r = 1 , so we have: 4 C 1 = 1 ! ( 4 − 1 )! 4 ! = 1 ! 3 ! 4 ! = ( 1 ) ( 3 × 2 × 1 ) 4 × 3 × 2 × 1 = 1 4 = 4 Therefore, there are 4 ways to choose the remaining person.
State the final answer The number of ways that both you and your friend are chosen is 4 C 1 = 4 .
Examples
Consider a scenario where a teacher needs to select a group of students for a project. If two specific students must be included in the group, this problem helps determine how many different groups the teacher can form by selecting the remaining members from the rest of the class. For example, if there are 6 students and a group of 3 needs to be formed, but 2 specific students must be in the group, then there are 4 ways to choose the remaining student.
