Using following numbers 10 17 18 19 20 21 23 26 28 29 30 33 35 how many sets of 6 numbers can we derive using each number only once

To solve the problem of how many sets of 6 numbers can be derived from the list of provided numbers (10, 17, 18, 19, 20, 21, 23, 26, 28, 29, 30, 33, 35), we will use the formula for combinations. The formula for combinations is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items to choose from, \( k \) is the number of items to choose, and \( ! \) denotes factorial. In this case: - The total number of items (n) is 13 (the total count of numbers you provided). - The number of items to choose (k) is 6. Now we can plug in the values: \[ C(13, 6) = \frac{13!}{6!(13-6)!} = \frac{13!}{6! \times 7!} \] Calculating the factorials, we can simplify: \[ C(13, 6) = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \] Calculating the numerator: - \( 13 \times 12 = 156 \) - \( 156 \times 11 = 1716 \) - \( 1716 \times 10 = 17160 \) - \( 17160 \times 9 = 154440 \) - \( 154440 \times 8 = 1235520 \) Now calculating the denominator: - \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \) Now, we divide the numerator by the denominator: \[ C(13, 6) = \frac{1235520}{720} = 1716 \] Thus, the total number of sets of 6 numbers that can be formed from the given numbers is: \[ \boxed{1716} \] Update (2025-08-04): To find out how many sets of 6 numbers can be derived from the given list of numbers \(10, 17, 18, 19, 20, 21, 23, 26, 28, 29, 30, 33, 35\) using each number only once, we can use the concept of combinations. The number of combinations of \(n\) items taken \(r\) at a time is given by the formula: \[ C(n, r) = \frac{n!}{r!(n - r)!} \] In this case: - \(n = 13\) (the total number of numbers) - \(r = 6\) (the number of numbers we want to choose) Now let's calculate the number of combinations: \[ C(13, 6) = \frac{13!}{6!(13 - 6)!} = \frac{13!}{6! \times 7!} \] Calculating the factorials: \[ C(13, 6) = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \] Calculating step by step: 1. Calculate the numerator: - \(13 \times 12 = 156\) - \(156 \times 11 = 1716\) - \(1716 \times 10 = 17160\) - \(17160 \times 9 = 154440\) - \(154440 \times 8 = 1235520\) 2. Calculate the denominator: - \(6! = 720\) 3. Compute the combination: \[ C(13, 6) = \frac{1235520}{720} = 1716 \] Thus, the number of sets of 6 numbers that can be derived from the given list is **1716**.