9. determine the number of four letter “words” that can be formed from the letters of the word produce if
a) each letter can only be used once

b) each letter can only be used once and the “word” must
i) contain only consonants
ii) begin and end with a consonant
iii) begin with a vowel
iv) contain the letter p
v) begin with d and end with a vowel
permutations and combinations lesson #1: the fundamental counting principle

Question

9. determine the number of four letter “words” that can be formed from the letters of the word produce if
a) each letter can only be used once

b) each letter can only be used once and the “word” must
  i) contain only consonants
  ii) begin and end with a consonant
  iii) begin with a vowel
  iv) contain the letter p
  v) begin with d and end with a vowel
permutations and combinations lesson #1: the fundamental counting principle

Accepted Answer

### Part a)
# Explanation:
## Step1: Identify the letters in "PRODUCE"
The word "PRODUCE" has 7 distinct letters: P, R, O, D, U, C, E.
## Step2: Apply the Fundamental Counting Principle for 4 - letter words (no repetition)
For the first letter, we have 7 choices. For the second letter, since one letter is already used, we have 6 choices. For the third letter, 5 choices remain. For the fourth letter, 4 choices remain.
Using the Fundamental Counting Principle, the number of 4 - letter words is $7\times6\times5\times4$
## Step3: Calculate the product
$7\times6 = 42$, $42\times5=210$, $210\times4 = 840$

# Answer: 840

### Part b) i)
# Explanation:
## Step1: Identify consonants in "PRODUCE"
The consonants in "PRODUCE" are P, R, D, C (since vowels are O, U, E). So there are 4 consonants.
## Step2: Apply the Fundamental Counting Principle for 4 - letter words (using only consonants, no repetition)
We need to form 4 - letter words using 4 consonants. For the first letter, we have 4 choices. For the second letter, 3 choices (since one consonant is used), for the third letter 2 choices, and for the fourth letter 1 choice.
Using the Fundamental Counting Principle, the number of words is $4\times3\times2\times1$
## Step3: Calculate the product
$4\times3 = 12$, $12\times2 = 24$, $24\times1=24$

# Answer: 24

### Part b) ii)
# Explanation:
## Step1: Identify consonants and vowels in "PRODUCE"
Consonants: P, R, D, C (4 consonants); Vowels: O, U, E (3 vowels)
## Step2: Analyze the positions (first and last are consonants, middle two can be any remaining letters)
- First letter (consonant): 4 choices.
- Last letter (consonant): After choosing the first consonant, we have 3 remaining consonants, so 3 choices.
- Middle two letters: After choosing first and last letters, we have $7 - 2=5$ remaining letters. For the second letter, 5 choices; for the third letter, 4 choices.
## Step3: Apply the Fundamental Counting Principle
The number of words is $4\times5\times4\times3$
## Step4: Calculate the product
$4\times5 = 20$, $20\times4=80$, $80\times3 = 240$

# Answer: 240

### Part b) iii)
# Explanation:
## Step1: Identify vowels in "PRODUCE"
Vowels in "PRODUCE" are O, U, E (3 vowels)
## Step2: Analyze the positions (first letter is a vowel)
- First letter (vowel): 3 choices.
- Remaining three letters: After choosing the first vowel, we have $7 - 1 = 6$ remaining letters. For the second letter, 6 choices; for the third letter, 5 choices; for the fourth letter, 4 choices.
## Step3: Apply the Fundamental Counting Principle
The number of words is $3\times6\times5\times4$
## Step4: Calculate the product
$3\times6=18$, $18\times5 = 90$, $90\times4=360$

# Answer: 360

### Part b) iv)
# Explanation:
## Step1: Analyze the position of letter P
Since the word must contain the letter P, we can think of it as: First, ensure that P is in one of the 4 positions, and then arrange the remaining 3 letters from the remaining 6 letters.
## Step2: Use the combination of choosing position for P and arranging other letters
- Number of ways to choose the position for P: There are 4 positions (since it's a 4 - letter word), so 4 choices.
- After placing P, we need to choose 3 letters from the remaining 6 letters and arrange them in the remaining 3 positions. The number of ways to choose and arrange 3 letters from 6 is $P(6,3)=\frac{6!}{(6 - 3)!}=6\times5\times4$
## Step3: Apply the Fundamental Counting Principle
The total number of words is $4\times(6\times5\times4)$
## Step4: Calculate the product
$6\times5\times4=120$, $4\times120 = 480$

# Answer: 480

### Part b) v)
# Explanation:
## Step1: Analyze the fixed positions (first is D, last is a vowel)
- First letter: Fixed as D, so 1 choice.
- Last letter (vowel): Vowels are O, U, E, so 3 choices.
- Middle two letters: After choosing D and the vowel, we have $7-2 = 5$ remaining letters. For the second letter, 5 choices; for the third letter, 4 choices.
## Step2: Apply the Fundamental Counting Principle
The number of words is $1\times5\times4\times3$
## Step3: Calculate the product
$5\times4=20$, $20\times3 = 60$

# Answer: 60

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QUESTION IMAGE

Question

Explanation:

Part a)

Step1: Identify the letters in "PRODUCE"

Step2: Apply the Fundamental Counting Principle for 4 - letter words (no repetition)

Step3: Calculate the product

Step1: Identify consonants in "PRODUCE"

Step2: Apply the Fundamental Counting Principle for 4 - letter words (using only consonants, no repetition)

Step3: Calculate the product

Step1: Identify consonants and vowels in "PRODUCE"

Step2: Analyze the positions (first and last are consonants, middle two can be any remaining letters)

Step3: Apply the Fundamental Counting Principle

Step4: Calculate the product

Answer:

Part b) i)