Permutations an...

$$\displaystyle ^{5}C_{4}=$$___

In how many ways can 8 books be distributed among 5 students if each student is eligible for any number of books?

In $$\displaystyle ^{3}C_{r}$$ the value of r can be ___

In how many ways can 3 diamond cards be drawn simultaneously from a pack of cards?

The diagram above shows the various paths along which a mouse can travel from point X, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from X to Y can the mouse take if it goes directly from X to Y without retracing any point along a path?

A researcher plans to identify each participant in a $$174$$ certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are $$12$$ participants,and each participant is to receive a different code?

A three-digit code for certain locks uses the digits $$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$$ according to the following constraints. The first digit cannot be $$0$$ or $$1$$, the second digit must be $$0$$ or $$1$$, and the second and third digits cannot both be $$0$$ in the same code. How many different codes are possible?

Four speakers will address a meeting where speaker $$Q$$ will always speak after $$P$$. Then, the number of ways in which the order of speakers can be prepared is

Mario's Pizza has 2 choices of crust: deep dish and thin-and-crispy. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. Finally, Mario's offers every pizza in extra cheese as well as regular. If Linda's volleyball team decides to order a pizza with 4 toppings, how many different choices do the teammates have at Mario's Pizza?

From the consecutive integers -10 to 10, inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?