Sheryl Jett wants to name her new baby so that his monogram (first, middle, and last initials) will be distinct letters in alphabetical order and he will share her last name. How many different monograms could she select?

Show that if u and v are solutions to the linear system Ax = b, then u - v is a solution to the associated homogeneous system Ax = 0.

Evaluate the expression $\frac{8!}{6!(8-6)!}$.

After rolling the first ball of a frame in a game of $10$-pin bowling, how many different pin configurations can remain (assuming all configurations are physically possible)?

Five thousand raffle tickets are sold. One first prize of $\$1000$, two second prizes of $\$500$ each, and three third prizes of $\$100$ each will be awarded, with all winners selected randomly. If the tickets were sold for $\$1$ each, how much profit goes to the raffle sponsor?

Determine the number of combinations (subsets) of $13$ things taken $6$ at a time.

Among the $2,598,960$ possible $5$-card poker hands from a standard $52$-card deck, how many contain at least one card that is not a heart (complement of "all hearts")?

What is the next row that would have the same property as rows $2$ and $4$ in the previous exercise?

A mining company operates two mines, each producing three grades of ore. The West Summit mine can produce 2 tons of low-grade ore, 3 tons of medium-grade ore, and 1 ton of high-grade ore in one hour of operation. The North Ridge mine can produce 2 tons of low-grade ore, 1 ton of medium-grade ore, and 2 tons of high-grade ore in one hour of operation. To satisfy existing orders, the company needs to produce at least 100 tons of low-grade ore, 60 tons of medium-grade ore, and 80 tons of high-grade ore. The cost of operating each mine varies, depending on conditions while extracting the ore. If it costs $400 per hour to operate the West Summit mine and$600 per hour to operate the North Ridge mine, how many hours should each mine operate to supply the required amounts of ore and minimize the cost of production? What is the minimum production cost?

Determine the number of combinations (subsets) of $9$ things taken $4$ at a time.

Determine the number of triangles (in any size) in the given figure.

Determine the number of triangles (in any size) in the figure.

If you roll two fair dice (say red and green), in how many ways can you obtain a $4$ on at least one of the dice?

A transit bus can travel in only two directions, north and east. From its starting point on the map shown, determine how many paths exist to reach the garage.