Answer :
To determine the number of distinct 10-letter words that can be formed from the given letters (W, V, W, Y, J, J, V, K, Z, O), we follow these steps:
1. Identify the total number of letters.
There are 10 letters given: W, V, W, Y, J, J, V, K, Z, and O.
2. Identify repeated letters.
- W appears 2 times.
- V appears 2 times.
- J appears 2 times.
- Y, K, Z, and O each appear 1 time.
3. Calculate the total number of permutations of 10 letters.
Without accounting for repetition, the total number of permutations of 10 distinct letters is calculated using the factorial function:
[tex]\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \][/tex]
4. Adjust for the repeated letters.
Since there are repetitions, we need to divide by the factorial of the number of occurrences of each repeated letter:
- For W (2 times): [tex]\( 2! = 2 \)[/tex]
- For V (2 times): [tex]\( 2! = 2 \)[/tex]
- For J (2 times): [tex]\( 2! = 2 \)[/tex]
The formula to correct for these repetitions is:
[tex]\[ \text{Total distinct permutations} = \frac{10!}{2! \times 2! \times 2!} \][/tex]
5. Calculate the denominator:
[tex]\[ 2! = 2 \quad \text{so} \quad 2! \times 2! \times 2! = 2 \times 2 \times 2 = 8 \][/tex]
6. Divide the total permutations by the product of the factorials of the repeated letters:
[tex]\[ \text{Total distinct permutations} = \frac{3,628,800}{8} = 453,600 \][/tex]
Thus, the number of distinct 10-letter words that can be formed from the given letters is 453,600.
1. Identify the total number of letters.
There are 10 letters given: W, V, W, Y, J, J, V, K, Z, and O.
2. Identify repeated letters.
- W appears 2 times.
- V appears 2 times.
- J appears 2 times.
- Y, K, Z, and O each appear 1 time.
3. Calculate the total number of permutations of 10 letters.
Without accounting for repetition, the total number of permutations of 10 distinct letters is calculated using the factorial function:
[tex]\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \][/tex]
4. Adjust for the repeated letters.
Since there are repetitions, we need to divide by the factorial of the number of occurrences of each repeated letter:
- For W (2 times): [tex]\( 2! = 2 \)[/tex]
- For V (2 times): [tex]\( 2! = 2 \)[/tex]
- For J (2 times): [tex]\( 2! = 2 \)[/tex]
The formula to correct for these repetitions is:
[tex]\[ \text{Total distinct permutations} = \frac{10!}{2! \times 2! \times 2!} \][/tex]
5. Calculate the denominator:
[tex]\[ 2! = 2 \quad \text{so} \quad 2! \times 2! \times 2! = 2 \times 2 \times 2 = 8 \][/tex]
6. Divide the total permutations by the product of the factorials of the repeated letters:
[tex]\[ \text{Total distinct permutations} = \frac{3,628,800}{8} = 453,600 \][/tex]
Thus, the number of distinct 10-letter words that can be formed from the given letters is 453,600.
