+4 In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers (1, 2, 3, 4, 5, 6, and 7). Among all the cards of each color, there is exactly one card labeled with each number. The cards in Ms. Q's deck are shown below.Yunseol draws five cards from Ms. Q's deck. What is the probability that exactly two cards have the same number?
+1418 We can solve this problem by considering the cases. There are four cases for the number of cards that have the same number: 2, 3, 4, and 5.
Case 1: 2 cards have the same number.
There are seven different numbers, so there are seven ways to choose which number will be shared by the two cards. Then, there are four ways to choose the color of the first card, and four ways to choose the color of the second card. However, we have double-counted the cases where the two cards are the same color, so we must divide by 2. Finally, there are four ways to choose the color of the third card, three ways to choose the color of the fourth card, and two ways to choose the color of the fifth card.
Therefore, the probability that exactly two cards have the same number is:
P(2 cards) = 7 * 4 * 4 / 2 * 4 * 3 * 2 = 840 / 5040 = 7/42
Case 2: 3 cards have the same number.
There are seven different numbers, so there are seven ways to choose which number will be shared by the three cards. Then, there are four ways to choose the color of the first card, four ways to choose the color of the second card, and three ways to choose the color of the third card. However, we have double-counted the cases where two of the cards are the same color, so we must divide by 3. Finally, there are three ways to choose the color of the fourth card, and two ways to choose the color of the fifth card.
Therefore, the probability that exactly three cards have the same number is:
P(3 cards) = 7 * 4 * 4 * 3 / 3 * 3 * 2 * 2 = 1920 / 5040 = 4/11
Case 3: 4 cards have the same number.
There are seven different numbers, so there are seven ways to choose which number will be shared by the four cards. Then, there are four ways to choose the color of the first card, four ways to choose the color of the second card, three ways to choose the color of the third card, and two ways to choose the color of the fourth card. However, we have double-counted the cases where two of the cards are the same color, so we must divide by 6. Finally, there is one way to choose the color of the fifth card.
Therefore, the probability that exactly four cards have the same number is:
P(4 cards) = 7 * 4 * 4 * 3 * 2 / 6 * 1 = 2520 / 5040 = 3/5
Case 4: 5 cards have the same number.
There are seven different numbers, so there are seven ways to choose which number will be shared by the five cards. Then, there are four ways to choose the color of the first card, four ways to choose the color of the second card, three ways to choose the color of the third card, two ways to choose the color of the fourth card, and one way to choose the color of the fifth card.
Therefore, the probability that exactly five cards have the same number is:
P(5 cards) = 7 * 4 * 4 * 3 * 2 / 1 = 1728 / 5040 = 2/3
Total probability:
To find the total probability that exactly two cards have the same number, we must add up the probabilities of each case:
P(exactly 2 cards) = P(2 cards) + P(3 cards) + P(4 cards) + P(5 cards)
= 7/42 + 4/11 + 3/5 + 2/3
= 77/210
