The numerator and denominator of a fraction are in the ratio of 3 to 5. If the numerator and denominator are both increased by 2, the fraction is now equal to 2/3
If n = the numerator and d = the denominator, which of the following systems of equations could be used
to solve the problem?

3n = 5d and 3n + 6 = 2d + 4

5n = 3d and 3n + 6 = 2d + 4

5n = 3d and 4n + 4 = 3d + 6

You are in a room of 22 people (including yourself). each of you has their own deck of 52 cards — assume each deck is well-shuffled, and can be assumed to be fair and independent of each other.1. each you draws the top card of your deck.what is the probability that at least one other person has drawn the same card as you? 2. let x be the number of people who drew a card that was also drawn by at least one other person. for example, if you all drew the same card, x=22, but if you all drew different cards, x=0. what is the expected value of x? 3. consider the same situation as the last question, but this time with 29 people: all 29 of you are in a room with your own decks of cards, and you draw one of them.what is the probability that at least one person draws the ace of spades? 4. let x be the number of the types of card drawn. for example, if everyone draws the same card, x=1, but if everyone draws a different card, x=29. what is the expected value of x?