Eleven graduate students have applied for two available teaching assistantships. In how many ways can these assistantships be awarded among the applicants if
(a) No preference is given to any one student?
(c) The group of applicants includes seven men and four women and it is stipulated that at least one woman must be awarded an assistantship?
(a) N = 11 choose 2 (=11C2)
(c) 11 choose 2 less 7 choose 2, i.e.
N minus number of ways of choosing two men.
(a) If no preference is given to any one student, the assistantships can be awarded in a combination of ways.
To calculate the number of ways, we can use the concept of combinations. The formula for combinations is given by
C(n, r) = n! / (r!(n-r)!),
where n is the total number of items and r is the number of items being chosen.
In this case, there are 11 students and 2 assistantships, so the number of ways to award the assistantships is given by C(11, 2).
Using the formula, we have:
C(11, 2) = 11! / (2!(11-2)!)
= 11! / (2!9!)
= (11 * 10 * 9!) / (2! * 9!)
= (11 * 10) / 2
= 55.
Therefore, there are 55 ways to award the assistantships when no preference is given to any one student.
(c) Now, let's consider the case where at least one woman must be awarded an assistantship.
We can break it down into two cases:
1) One woman is awarded an assistantship, and the other assistantship can go to any of the remaining 10 students.
2) Both assistantships are awarded to women.
In the first case, one woman must be chosen from the 4 women, and then one student must be chosen from the remaining 10 students, which includes 3 women and 7 men. The number of ways to choose is given by C(4, 1) * C(10, 1) = 4 * 10 = 40.
In the second case, both assistantships can be awarded to any of the 4 women. So the number of ways is C(4, 2) = 6.
To calculate the total number of ways, we sum the two cases:
Total ways = Case 1 + Case 2
= 40 + 6
= 46.
Therefore, there are 46 ways to award the assistantships when at least one woman must be awarded an assistantship.
To solve this problem, we need to consider different scenarios based on the given conditions.
a) No preference is given to any one student:
In this case, we need to find all possible ways to award the assistantships without any restrictions. This means that each assistantship can be awarded to any of the eleven graduate students.
To calculate the number of ways, we take into account that each assistantship has two options: either an applicant gets it or does not get it (since there are only two available). So, for each student, we multiply their options, resulting in:
Number of ways = 2 * 2 * 2 * ... (11 times) = 2^11 = 2048.
Therefore, there are 2048 ways to award the assistantships when no preference is given to any one student.
c) At least one woman must be awarded an assistantship:
In this case, we have to make sure that at least one of the four women among the applicants receives an assistantship.
To calculate the number of ways, we need to consider the possibilities when:
1. Exactly one woman receives an assistantship.
2. Exactly two women receive an assistantship.
3. Exactly three women receive an assistantship.
4. All four women receive an assistantship.
Let's calculate each possibility separately:
1. Exactly one woman receives an assistantship:
There are four women and seven men, so we can choose one woman from the four and assign the remaining assistantship to any of the 11 students.
Number of ways = 4C1 * 11 = 44.
2. Exactly two women receive an assistantship:
We can choose two women from the four and assign the remaining assistantship to any of the 11 students.
Number of ways = 4C2 * 11 = 6 * 11 = 66.
3. Exactly three women receive an assistantship:
We can choose three women from the four and assign the remaining assistantship to any of the 11 students.
Number of ways = 4C3 * 11 = 4 * 11 = 44.
4. All four women receive an assistantship:
There is only one possible combination when all four women are awarded the assistantships.
Number of ways = 1.
Now, we sum up the number of ways for each possibility:
Total number of ways = (number of ways for exactly one woman) + (number of ways for exactly two women) + (number of ways for exactly three women) + (number of ways for all four women)
= 44 + 66 + 44 + 1
= 155.
Therefore, there are 155 ways to award the assistantships when at least one woman must be awarded an assistantship.