Ten 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?
(b) One particular student must be awarded an assistantship?
(c) The group of applicants includes six men and four women and it is stipulated that at least one woman must be awarded an assistantship?
(a) No preference is given to any one student:
Since there are two available assistantships and no preference is given, each assistantship can be awarded to any of the ten graduate students. Therefore, for each assistantship, there are ten possible choices. Hence, the total number of ways the assistantships can be awarded is 10 * 10 = 100.
(b) One particular student must be awarded an assistantship:
If one particular student must be awarded an assistantship, then we can treat this student as already receiving one of the assistantships. So, there is only one assistantship left to be awarded to the remaining nine graduate students. Therefore, there are 9 possible choices for the second assistantship. Hence, the total number of ways the assistantships can be awarded in this case is 1 * 9 = 9.
(c) At least one woman must be awarded an assistantship:
To calculate the number of ways to award the assistantships, we can consider two cases:
Case 1: One woman is awarded both assistantships.
In this case, there are 4 choices for the woman who receives both assistantships, and there are no choices for the remaining graduate students. So, there is only 1 way to award the assistantships.
Case 2: One woman is awarded one assistantship, and the other assistantship is awarded to anyone (either a man or another woman).
In this case, there are 4 choices for the woman who receives one assistantship. The remaining assistantship can be awarded to any of the remaining nine graduate students (six men and three women). So, there are 9 choices for the second assistantship. Therefore, there are 4 * 9 = 36 ways to award the assistantships in this case.
Adding the number of ways from both cases, the total number of ways to award the assistantships is 1 + 36 = 37.