Possible Solutions: Individual Portion of Course Exam
MAT 305 Summer 1997

1.
a) 13 gloves
The worst case scenario here is that Tim grabs twelve gloves all of the same hand. A 13th glove has to pair up with one of the first twelve.

b) 13 gloves
The worst case again is to get all the gloves of one hand (left or right) before getting one of the opposite hand. The 13th glove will not only pair up with one of the first 12 to create a matched pair, it will also match in color.

2.
a) C(5,3) = C(5,2)
Select two of the five newborns for tag switching, or, equivalently, three of the newborns for correct tags. Order is inconsequential.

b) C(3,1)*C(2,1)
We now select one of each gender for switching tags.

c)

3.
a) C(8,3)*3! = P(8,3)
Remove (at least in your mind) three of the ten seats. Imagine the eight spaces within and among the seven remaining seats. Select 3 of these eight spaces and place the grouchy folks in them. They can permute themselves 3! ways for each of the selected set of three seats, so multiply C(8,3) by 3!. Note that we stipulate nothing about whether the remaining seats are occupied and hence are not concerned with how many ways any people in the seven other seats may be arranged.

b) C(n - k + 1, k)*k! = P(n - k + 1, k)
Remove k seats, consider the (n - k + 1) spaces among and within the remaining (n - k) seats. Choose k of those seats and permute their occupants. We must have n-k+1=>k, or, equivalently, 2k-1<=n.

4. 28 paths
Count the paths from M outward just as you might have counted paths in the street-grid problems, moving from letter to letter. Sum the values you get for each of the 12 Hs on the outer edge of the diagram.

5.
a) P(9,6)

b) C(11,3)
We solve the equation x1 + x2 + x3 + x4 = 16 under the restriction that each x(i) be a positive integer 2 or greater.

6.
a) 174
Use either a Venn diagram or the inclusion-exclusion principle. The values given on the page can be directly substituted into a form of the IEP:

1400 - (675 + 682 + 684) + (195 + 467 + 318) - 165.

A Venn diagram is also shown here.

b) 30
Pick out the particular subset within the Venn diagram.

c) 508
Sum 174 and 334, the two values NOT in the married nor the female sets.