Consider the game of Nim (with normal play rules): (a) Consider piles position (3, 7) (i.e., two piles with the first pile having 3 and the second having 7 chips). Does this represent a P-position or N-position? Describe the winning strategy. (b) Consider piles position (5, 0, 3, 2). Does this represent a P-position or N-position and why? You don't need to describe the winning strategy

At a back-to-school party, one of your friends lets you play a two-stage game where you pay $3 to play. The game works as follows: flip a fair coin, noting the side showing, and roll a fair standard 4-sided die (numbered 1–4), noting the number showing. If the die shows a 3 and the coin shows tails, then you win $22. If the coin shows heads and the die shows an even number, then you win $9. Otherwise, you do not win anything. Let X be your net winnings. (a) Create a probability distribution for X. Enter the possible values of X in ascending order from left to right. All probabilities should be exact. X       P(X)       (b) Compute your expected net winnings for the game. Round your answer to the nearest cent.

Toward the end of a game of Scrabble you hold the letters D,O,G and Q.  You can choose three of these four letters and arrange them in order in how many different ways?

In part (a) what is the largest number of caramels that Brenda ca you play your strategy guaranteeing the peppermint patty? 6. Along the coast there are 3 lighthouses, whose lights can be seen blinking at night by sailors near the shore. The first light shines for 3 seconds then is off for 3 seconds and repeats. The second light shines for 4 seconds then is off for 4 seconds and repeats. The third light shines for 5 seconds then is off for 5 seconds and repeats. All three lights have just come on together. (a) When is the first time the three lights will all be off at the same time? (b) When is the next time all three lights will come on at the same moment? (c) For a sailor off-shore, how many "light-on" blinks will she see between the two times all three come on together? Note: Parts (a) and (b) of this problem are found in the homework of school children ages 7-11 in the U.K. In May, 2018 the Daily Mail reported on this problem: "Ridicu- lous maths problem intended for PRIMARY…