Question: A committee of 7 members is to be formed to put up the Christmas decorations at Hogwarts. The following volunteered: 9 teachers, 7 Hufflepuff students, 4 ghosts, 6 Gryffindors, and 3 Prefects. How many possible committees can be formed by Professor Dumbledore if a committee must have only 1 Prefect, only 1 ghost, and at most 2 teachers?

1. 123,552
2. 200,772
3. 216,216
4. 316,188
5. 1,560,780

Correct Answer: (C)
Approach Solution : 1

There are three cases that can make up the committee.

1) One prefect, one ghost, and five students;
2) One prefect, one ghost, one teacher, and four students; or
3) One prefect, one ghost, two teachers, and three students can make up the committee.

Let's figure out how many committees there will be for each of the cases.

Case - 1 : 5 students, 1 ghost, and 1 prefect.
You will see that the total number of students are 7+6 = 13.
The number of ways to select 5 students from a group of 13 students can be calculated using the formula, 13C5 = 13!/(5!*8!) = (13*12*11*10*9)/(5*4*3*2) = 13 * 11 * 9 = 1287 ways.
There are 3*4 *1,287 = 15,444 different ways to assemble the committee of one prefect, one ghost, and five students since there are three options for the prefect and four options for the ghost.

Case - 2 : 4 students, 1 teacher, 1 ghost, and 1 prefect :

The number of ways to select 4 students from a group of 13 students is 13C4 = 13!/(4!*9!) = (13*12*11*10)/(4*3*2) = 13 * 11 * 5 = 715 ways.
There are 3 * 4 * 9 * 715 = 77,220 ways to assemble the committee of one prefect, one ghost, one teacher, and four students in addition to the 3 preferences for the prefect, 4 preferences for the ghost, and 9 preferences for the teacher.

Case - 3 : 3 students, 2 teachers, 1 ghost, and 1 prefect

The number of ways to select 3 students from a group of 13 students is 13C3 = 13!/(3!*10!) = (13*12*11)/(3*2) = 13 * 2 * 11 = 286 ways.
In 9C2 = 9!/(2!*7!) = (9*8)/2 = 36 ways, two teachers can be selected from a group of nine teachers.
Together, there are 3 * 4 * 36 * 286 =123,552 different ways to assemble the committee of one prefect, one ghost, two teachers, and three students.
The number of ways we can form the committee can be calculated by adding the number of ways for each case.
=> 15,444 + 77,220 + 123,552 = 216,216

Approach Solution : 2

There are nine teachers, seven Hufflepuff students, four ghosts, six Gryffindors, and three Prefects.
The selection methods are given below.

1) Only one prefect will have three options
2) Only 1 ghost will have 4 options

There are now only 5 members left.

3) No more than two teachers

a) no teacher = 13C5 = (13∗12∗11∗10∗9) / (5∗4∗3∗2) = 13∗11∗9
b) 1 teacher = 13C4∗9C1 = (13∗12∗11∗10) / (4!∗9) = 13∗11∗5∗9
c) 2 teachers = 13C3∗9C2 = [(13∗12∗11)/3!] ∗[(9∗8)/2)] =13∗11∗9∗8

Now we can write, (13∗11∗9) + (13∗11∗9∗5) + (13∗11∗9∗8) = (13∗11∗9) * (1+5+8) = 13∗11∗9∗14
Therefore the total number of ways is
=> 13∗11∗9∗14∗3∗4 = 216,216

“A committee of 7 members is to be formed to put up the Christmas” - is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

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