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Chapter 20: Problem 40

In how many ways can 4 consonants be chosen from the letters of the word SOMETHING? (1) \({ }^{9} \mathrm{C}\), (2) \({ }^{6} \mathrm{C}\) (3) \({ }^{4} \mathrm{C}\) (4) \({ }^{4} \mathrm{C}\)

Short Answer

Expert verified
Answer: 5 ways.

Step by step solution

01

(Step 1: Identifying the consonants)

: In the word SOMETHING, the consonants are S, M, T, N, and G. So, there are a total of 5 consonants.
02

(Step 2: Calculating the number of ways)

: To calculate the number of ways to choose 4 consonants out of 5, we will use the combination formula which is given by: nCr = n! / (r!(n-r)!) Where n is the total number of options (in our case, the total number of consonants) and r is the number of choices we want to make (in our case, choosing 4 consonants). Here, the factorial (!) represents the product of all positive integers up to that number. In our case, n = 5 and r = 4. So, the formula will look like: 5C4 = 5! / (4!(5-4)!)
03

(Step 3: Calculating the factorials)

: Now, we need to calculate the factorials in the formula: 5! = 5 脳 4 脳 3 脳 2 脳 1 = 120 4! = 4 脳 3 脳 2 脳 1 = 24 (5-4)! = 1! = 1
04

(Step 4: Applying the factorials to the combination formula)

: Now, plug the factorial values back into the combination formula: 5C4 = 120 / (24 脳 1) = 5 So, there are 5 different ways to choose 4 consonants out of the 5 available consonants in the word SOMETHING. The correct answer is: (2) \({ }^{6} \mathrm{C}\)

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