How many ways can 10 different items be selected from a group of 15? (2023)

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Question: How many ways can 10 different items be selected from a group of 15? (2023)

Options:

Correct Answer: 3003

Exam Year: 2023

Solution:

The number of ways to choose 10 items from 15 is given by 15C10 = 15C5 = 3003.

How many ways can 10 different items be selected from a group of 15? (2023)

How many ways can 10 different items be selected from a group of 15? (2023)

Step 1: Understand that we want to choose 10 items from a total of 15 different items.
Step 2: Recognize that the number of ways to choose items is calculated using combinations, denoted as nCr, where n is the total number of items and r is the number of items to choose.
Step 3: In this case, we need to calculate 15C10, which means choosing 10 items from 15.
Step 4: Use the combination formula: nCr = n! / (r! * (n - r)!), where '!' denotes factorial (the product of all positive integers up to that number).
Step 5: Plug in the values: 15C10 = 15! / (10! * (15 - 10)!) = 15! / (10! * 5!).
Step 6: Simplify the calculation: 15C10 = 15! / (10! * 5!) = (15 × 14 × 13 × 12 × 11) / (5 × 4 × 3 × 2 × 1).
Step 7: Calculate the numerator: 15 × 14 × 13 × 12 × 11 = 360360.
Step 8: Calculate the denominator: 5 × 4 × 3 × 2 × 1 = 120.
Step 9: Divide the numerator by the denominator: 360360 / 120 = 3003.
Step 10: Conclude that there are 3003 different ways to select 10 items from a group of 15.

Combinatorics – The question tests the understanding of combinations, specifically how to calculate the number of ways to choose a subset of items from a larger set.
Binomial Coefficient – The use of the binomial coefficient notation (nCr) to represent the number of combinations is a key concept being tested.