Explore essential concepts of permutations, combinations, and probability with this engaging quiz designed to enhance your problem-solving skills. Master the principles of counting methods and probability calculations through real-life scenarios and practical applications.
In how many different ways can 4 distinct books be arranged on a shelf?
Explanation: The number of ways to arrange 4 distinct items is given by 4 factorial, which is 4 × 3 × 2 × 1 = 24. Option 16 is incorrect as it confuses permutations with binary choices. Option 12 is incorrect because it omits one stage of multiplication. Option 8 relates to the number of ways of arranging 3 items only. Only 24 accurately accounts for all possible orders of 4 unique books.
A club of 10 members needs to choose a committee of 3. How many ways can the committee be formed if the order does not matter?
Explanation: The number of ways to choose 3 out of 10 without regard to order is a combination: 10C3 = 120. Option 720 is 6! and may confuse with arrangement, not selection. Option 30 is 10P2, which calculates permutations of 2 instead of 3. Option 60 incorrectly divides rather than uses the combination formula. Thus, 120 is correct because it uses combinations for unordered selection.
If two fair coins are tossed simultaneously, what is the probability of getting exactly one head?
Explanation: There are four possible outcomes: HH, HT, TH, and TT. Exactly one head occurs in two out of four cases (HT and TH), so the probability is 2/4 = 1/2. Option 1/4 is only correct if one outcome is favorable, which is not the case here. Option 1/3 and 3/4 are incorrect representations of probability in this scenario. Therefore, 1/2 properly reflects the chance of exactly one head.
How many 3-letter passwords can be formed using the letters A, B, C, D, and E, if no letter can be used more than once?
Explanation: Since repetition is not allowed, this is a permutation: 5 × 4 × 3 = 60 ways. Option 125 counts with repetition (5 × 5 × 5). Option 27 is 3³, also allowing repetition for only three letters. Option 10 calculates combinations without order. Sixty is the only accurate count for ordered, non-repeating arrangements.
What is the probability of drawing a red card from a standard deck of 52 playing cards?
Explanation: There are 26 red cards (hearts and diamonds) in a 52-card deck, so the probability is 26/52, which simplifies to 1/2. Option 1/4 would be correct if there were only 13 red cards, but there are 26. Option 1/13 misrepresents the ratio among suits, not colors. Option 3/4 grossly exaggerates the number of red cards. Therefore, 1/2 is the precise probability.