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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Hey Princess, good to see you again. Here are the solutions to the exercises: 1) To find the number of different ways the letters in the word "BOOK" can be arranged, we use the formula for permutations with repetitions. Step 1: Identify the total number of letters and any repeated letters. The word "BOOK" has 4 letters. The letter 'O' is repeated 2 times. Step 2: Apply the permutation formula for repeated items. Number of ways = (n!)/(n_1!) where n is the total number of letters and n_1 is the number of times a letter is repeated. Number of ways = (4!)/(2!) = (4 × 3 × 2 × 1)/(2 × 1) = (24)/(2) = 12 The given answer is 12 ways. The answer is 12 ways. 2) To find the number of ways 5 books can be arranged on a shelf, we calculate the permutations of 5 distinct items. Step 1: Identify the number of items to be arranged. There are 5 books. Step 2: Calculate the factorial of the number of items. Number of ways = 5! = 5 × 4 × 3 × 2 × 1 = 120 The given answer is 120 ways. The answer is 120 ways. 3) To find the number of different seating arrangements for 8 people in a row, we calculate the permutations of 8 distinct items. Step 1: Identify the number of people to be arranged. There are 8 people. Step 2: Calculate the factorial of the number of people. Number of ways = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320 The given answer is 40,320. The answer is 40,320 ways. 4) To find the number of ways the letters in "MANGO" can be arranged if the vowels must always come together, we treat the vowels as a single unit. Step 1: Identify vowels and consonants and group the vowels. The word "MANGO" has letters M, A, N, G, O. Vowels: A, O. Consonants: M, N, G. Treat (AO) as one unit. Now we are arranging M, N, G, (AO). This is 4 units. Step 2: Arrange the units. The 4 units can be arranged in 4! ways. 4! = 4 × 3 × 2 × 1 = 24 Step 3: Arrange the letters within the vowel unit. The vowels (A, O) can be arranged among themselves in 2! ways. 2! = 2 × 1 = 2 Step 4: Multiply the arrangements from Step 2 and Step 3. Total ways = 24 × 2 = 48 The given answer is 48 ways. The answer is 48 ways. 5) To find the number of ways the letters in "ENGINEER" can be arranged such that all vowels are together, we treat the vowels as a single unit and account for repetitions. Step 1: Identify vowels and consonants and group the vowels. The word "ENGINEER" has 8 letters: E, N, G, I, N, E, E, R. Vowels: E, I, E, E (3 E's, 1 I). Consonants: N, G, N, R (2 N's, 1 G, 1 R). Group the vowels: (EIE E). Treat this as one unit. Now we are arranging the units: (EIE E), N, G, N, R. This is 5 units. Step 2: Arrange the units, accounting for repeated consonants. The units are (Vowel Group), N, G, N, R. There are 5 units in total, with 'N' repeated 2 times. Arrangement of units = (5!)/(2!) = (5 × 4 × 3 × 2 × 1)/(2 × 1) = (120)/(2) = 60 Step 3: Arrange the letters within the vowel unit, accounting for repetitions. The vowel group is (EIE E). There are 4 vowels, with 'E' repeated 3 times. Arrangement of vowels = (4!)/(3!) = (4 × 3 × 2 × 1)/(3 × 2 × 1) = 4 Step 4: Multiply the arrangements from Step 2 and Step 3. Total ways = 60 × 4 = 240 The given answer is 120 ways. My calculation gives 240 ways. Let me recheck. Recheck for Question 5: Word: ENGINEER Total letters: 8 Vowels: E, I, E, E (4 vowels, 3 E's, 1 I) Consonants: N, G, N, R (4 consonants, 2 N's, 1 G, 1 R) Treat all vowels as one block: (EIEE). Now we have 5 "items" to arrange: (EIEE), N, G, N, R. Number of ways to arrange these 5 items: There are 5 items, but 'N' is repeated twice. (5!)/(2!) = (120)/(2) = 60 Number of ways to arrange the letters within the vowel block (EIEE): There are 4 letters, with 'E' repeated 3 times. (4!)/(3!) = (24)/(6) = 4 Total number of arrangements = (Arrangement of blocks) × (Arrangement within vowel block) Total ways = 60 × 4 = 240 The provided answer of 120 ways is incorrect based on my calculation. The correct answer is 240 ways. The answer is 240 ways. 6) To find the number of different ways 4 paintings can be arranged on a wall, we calculate the permutations of 4 distinct items. Step 1: Identify the number of items to be arranged. There are 4 paintings. Step 2: Calculate the factorial of the number of items. Number of ways = 4! = 4 × 3 × 2 × 1 = 24 The given answer is 24 ways. The answer is 24 ways. Send me the next one 📸