Came across this question on this site itself and I am struggling to get the answer the way they describe it and why my approach is at fault
here is the question:
EXAMPLE 10 A three-person committee must be chosen from a group of 7 professors and 10 graduate students. If at least one of the people on the committee must be a professor, how many different groups of people could be chosen for the committee?
My approach :
Since we need to have a professor for the 3-member committee , we can choose the first person as professor in 7 ways .
Rest two positions can be filled with remaining professors or students in (16 C 2 )
total number of combinations for filling 3 member committe -
first position with professor * rest with 2 from 16 mombers ( 6 professors + 10 students) is
7 * (16 C 2) = 7 * (16 * 15 )/2
but what is baffling is total number of combinations for choosing 3 members from 17 ( 7 professors + 10 students) without restriction of atleast one professor in the committie = 17 C 2 = 680
how can this combination be less than restricted combination
can someone explain me where my thought process took a wrong turn ..
Here is how I solved this problem...
There are three cases in which you have a professor that is part of the comitte:
1 prof, 2 students
2 prof, 1 students
3 prof, 0 students
Here are the total number of combinations I found for each case:
1 prof, 2 students: 7*10*9 = 630
2 prof, 1 students: 7*6*10 = 420
3 prof, 0 students: 7*6*5 = 210
So then I got 1260 total (630+420+210) which is D.