Jackson has 20 books, but has space for only 14 on his shelf. He can't decide which books to place on the shelf, so he puts them all in a box, reaches in and randomly selects 14 of them to go on the shelf. a. Is this a combination or a permutation? Why? b. How many ways can 14 books be selected out of 20 in this manner? a. This is a because order

a. This is a combination because the order in which the books are selected does not matter. The combination formula is used when the order of selection does not affect the outcome. b. The number of ways to select 14 books out of 20 can be calculated using the combination formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where: - \( n \) is the total number of items (20 books in this case), - \( k \) is the number of items to be selected (14 books in this case). Substitute the values into the formula to find the number of ways to select 14 books out of 20. Simplify the expression by following steps: - step0: Solution: \(\frac{20!}{\left(14!\times \left(20-14\right)!\right)}\) - step1: Remove the parentheses: \(\frac{20!}{14!\times \left(20-14\right)!}\) - step2: Subtract the numbers: \(\frac{20!}{14!\times 6!}\) - step3: Expand the expression: \(\frac{20\times 19\times 18\times 17\times 16\times 15\times 14!}{14!\times 6!}\) - step4: Simplify: \(\frac{20\times 19\times 18\times 17\times 16\times 15}{6!}\) - step5: Multiply the terms: \(\frac{27907200}{6!}\) - step6: Calculate the value: \(\frac{27907200}{720}\) - step7: Reduce the fraction: \(38760\) The number of ways to select 14 books out of 20 is 38760.