Assume that the student has a cup with 14 writing implements: 7 pencils, 5 ball point pens, and 2 felt-tip pens.
How many such subcommittees are possible if each subcommittee must contain at least 1 and no more than 2 Republicans?
In how many ways can the selection be made if no more than one ball point pen is selected?
this is the real question not the other one!!
To find the number of possible subcommittees with the given conditions, we can use the concept of combinations.
First, let's identify the number of Republicans in the group. We know that there are a total of 14 writing implements, consisting of 7 pencils, 5 ballpoint pens, and 2 felt-tip pens. Since no more than 2 Republicans can be in a subcommittee, we have two cases to consider:
Case 1: There is one Republican in the subcommittee.
In this case, we need to choose 1 Republican and select the remaining members from the non-Republican writing implements. We can choose the Republican in 2 ways (since there are only 2 felt-tip pens). For the non-Republican members, we can select any 2 writing implements out of the remaining 12 (7 pencils and 5 ballpoint pens). This can be done in C(12, 2) ways.
Case 2: There are two Republicans in the subcommittee.
Here, we need to choose 2 Republicans and select the remaining members from the non-Republican writing implements. We can choose the 2 Republicans in C(2, 2) = 1 way (since there are only 2 felt-tip pens). For the non-Republican members, we can select any 1 writing implement out of the remaining 12 (7 pencils and 5 ballpoint pens). This can be done in C(12, 1) ways.
To get the total count of subcommittees, we need to add the results from both cases:
Total subcommittees = (Number of subcommittees with 1 Republican) + (Number of subcommittees with 2 Republicans)
= C(2, 1) * C(12, 2) + C(2, 2) * C(12, 1)
Now, we can calculate the values and find the answer:
Total subcommittees = (2) * (C(12, 2)) + (1) * (C(12, 1))
= 2 * (12! / (2! * (12-2)!)) + 1 * (12! / (1! * (12-1)!))
Simplifying further:
Total subcommittees = 2 * (12 * 11 / (2 * 1)) + 1 * 12
= 12 * 11 + 12
= 132 + 12
= 144
Therefore, there are 144 possible subcommittees that satisfy the given conditions.