If 10⁵⁰ - 74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?

The function f is defined for each positive three-digit integer n by f(n) = 2^x 3^y 5^Z, where x, y, and z are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f(m) = 9f(v), then m -v =

If x is to be chosen at random from the set {1, 2, 3, 4} and y is to be chosen at random from the set {5, 6, 7}, what is the probability that xy will be even?

M and N are integers such that 6 < M < N. What is the value of N ?

(1) The greatest common divisor of M and N is 6.

(2) The least common multiple of M and N is 36.

There are 10 books on a shelf, of which 4 are paperbacks and 6 are hardbacks. How many possible selections of 5 books from the shelf contain at least one paperback and at least one hardback?

There are 5 cars to be displayed in 5 parking spaces, with all the cars facing the same direction. Of the 5 cars, 3 are red, 1 is blue, and 1 is yellow. If the cars are identical except for color, how many different display arrangements of the 5 cars are possible?

The subsets of the set {w, x, y} are {w}, {x}, {y}, {w, x}, {w, y}, {x, y}, {w, x, y}, and { } (the empty subset). How many subsets of the set {w, x, y, z} contain w ?

Stations X and Y are connected by two separate, straight, parallel rail lines that are 250 miles long. Train P and train Q simultaneously left Station X and Station Y, respectively, and each train traveled to the other's point of departure. The two trains passed each other after traveling for 2 hours. When the two trains passed, which train was nearer to its destination?

(1) At the time when the two trains passed, train P had averaged a speed of 70 miles per hour.

(2) Train Q averaged a speed of 55 miles per hour for the entire trip.

A survey of employers found that during 1993 employment costs rose 3.5 percent, where employment costs consist of salary costs and fringe- benefit costs. If salary costs rose 3 percent and fringe-benefit costs rose 5.5 percent during 1993, then fringe-benefit costs represented what percent of employment costs at the beginning of 1993 ?

In the xy-plane shown, the shaded region consists of all points that lie above the graph of~$ y = x^2 - 4x$~ and below the x-axis. Does the point (a,b) (not shown) lie in the shaded region if b < 0 ?

(1) 0 < a < 4

(2) ~$ a^2 - 4a < b$~