Question

1) A man has 2 pennies, 3 nickels, 1 dime, and 2 quarters. How many different sums of money can he make using one or more of these 8 coins?
a) 8
b) 12
c) 47
d) 77

2) A train left Albany for Buffalo, a distance of 290 miles, at 10:10 am. The train was scheduled to reach Buffalo at 3:45 pm. If the average rate of the train on this trip was 50 miles per hour, it arrived in Buffalo
a) about 5 minutes ahead of schedule
b) on time
c) about 5 minutes late
d) about 13 minutes later
3) more than a quarter of an hour late

Answer 1

He can use the 2 pennies in 3 ways, take none, take 1 or take 2
He can use the 3 nickels in 4 ways, take none, take 1, etc
the dime in 2 ways,
and the quarters in 3 ways

total number of ways = (3)(4)(2)(3) = 72
but that includes taking none of any coins,

so number of ways to use the coins is 71

Here is a little silly computer program that finds and lists all the possible sumes

10 for p = 0 to 2
20 for n = 0 to 3
30 for d = 0 to 1
40 for q = 0 to 2
50 sum = p+5*n+10*d+25*q
60 print sum,
70 next q
80 next d
90 next n
100 next p
>run
0 25 50 10 35 60 5 30 55 15 40 65 10 35 60 20
45 70 15 40 65 25 50 75 1 26 51 11 36 61 6 31
56 16 41 66 11 36 61 21 46 71 16 41 66 26 51 76
2 27 52 12 37 62 7 32 57 17 42 67 12 37 62 22
47 72 17 42 67 27 52 77

Notice there are duplications,
e.g. picking 2 nickels or 1 dime yields the same sum

All you have to do is eliminate all duplications, and not count the 0
I think there are 47 or them