This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Right Nqøbizithå🤍🩺, let's solve this probability problem about arranging books. First, let's determine the total number of books and the total number of ways to arrange them. • Mathematics books: 4 • History books: 3 • Science books: 3 • Biology books: 2 Total number of books = 4 + 3 + 3 + 2 = 12 books. Assuming all books are distinct (e.g., different titles even if of the same subject), the total number of ways to arrange 12 books on a shelf is 12!. N = 12! = 479,001,600 23.1 Determine the probability that all books of the same subject land up next to each other. Step 1: Treat each subject group as a single block. We have 4 blocks: (4 Math books), (3 History books), (3 Science books), (2 Biology books). Step 2: Arrange these 4 blocks. The number of ways to arrange these 4 distinct blocks is 4!. 4! = 4 × 3 × 2 × 1 = 24 Step 3: Arrange the books within each block. • The 4 Mathematics books can be arranged in 4! ways (4 × 3 × 2 × 1 = 24). • The 3 History books can be arranged in 3! ways (3 × 2 × 1 = 6). • The 3 Science books can be arranged in 3! ways (3 × 2 × 1 = 6). • The 2 Biology books can be arranged in 2! ways (2 × 1 = 2). Step 4: Calculate the total number of favorable arrangements (E_1). E_1 = (arrangements of blocks) × (arrangements within M) × (arrangements within H) × (arrangements within S) × (arrangements within B) E_1 = 4! × 4! × 3! × 3! × 2! E_1 = 24 × 24 × 6 × 6 × 2 = 41,472 Step 5: Calculate the probability. P(E_1) = (E_1)/(N) = (41,472)/(479,001,600) P(E_1) = (1)/(11,550) The probability that all books of the same subject land up next to each other is (1)/(11,550). 23.2 Determine the probability that just the History books will be together. Step 1: Treat the 3 History books as a single block. The other books are 4 Mathematics, 3 Science, and 2 Biology books. These are 4+3+2=9 individual books. Step 2: Arrange the block of History books and the 9 other individual books. We are arranging a total of 1 (History block) + 9 (other books) = 10 items. The number of ways to arrange these 10 items is 10!. 10! = 3,628,800 Step 3: Arrange the books within the History block. The 3 History books within their block can be arranged in 3! ways. 3! = 3 × 2 × 1 = 6 Step 4: Calculate the total number of favorable arrangements (E_2). E_2 = (arrangements of 10 items) × (arrangements within History block) E_2 = 10! × 3! E_2 = 3,628,800 × 6 = 21,772,800 Step 5: Calculate the probability. P(E_2) = (E_2)/(N) = (10! × 3!)/(12!) We can simplify this expression: P(E_2) = (10! × 3!)/(12 × 11 × 10!) P(E_2) = (3!)/(12 × 11) P(E_2) = (6)/(132) P(E_2) = (1)/(22) The probability that just the History books will be together is (1)/(22). 3 done, 2 left today. You're making progress.