A certain university will select 1 of 7 candidates eligible to fill a position in the mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the computer science department. If none of the candidates is eligible for a position in both departments, how many different sets of 3 candidates are there to fill the 3 positions?

The points R, T, and U lie on a circle that has radius 4.

If the length of arc RTU is~$4π \over 3$~ ,what is the length of line segment RU ?

~$2x+y=12$~
~$\left | y \right |\leq 12$~

For how many ordered pairs (x,y) that are solutions of the system above are x and y both integers?

When the figure above is cut along the solid lines, folded along the dashed lines, and taped along the solid lines, the result is a model of a geometric solid. This geometric solid consists of 2 pyramids, each with a square base that they share. What is the sum of the number of edges and the number of faces of this geometric solid?

If the variables X, Y, and Z take on only the values 10, 20, 30, 40, 50, 60, or 70 with frequencies indicated by the shaded regions above, for which of the frequency

distributions is the mean equal to the median?

               ~$3,k,2,8,m,3$~

The arithmetic mean of the list of numbers above is 4. If k and m are integers and k≠m ,what is the median of the list?

In pentagon PQRST, PQ = 3, QR = 2, RS = 4, and ST = 5. Which of the lengths 5, 10, and 15 could be the value of PT ?

For every even positive integer m, f(m) represents the product of all even integers from 2 to m, inclusive. For example, f (12) = 2 x 4 x 6 x 8 x 10 x 12. What is the greatest prime factor of f(24) ?

If x and k are integers and (12^x)(4^2x + ¹) = (2^k)(3²), what is the value of k ?

On a certain transatlantic crossing, 20 percent of a ship's passengers held round-trip tickets and also took their cars aboard the ship. If 60 percent of the passengers with round-trip tickets did not take their cars aboard the ship, what percent of the ship's passengers held round-trip tickets?