The number of ways can the letters of the word FORECAST taken 3 at a time and the word MILKY taken 2 at a time be arranged are:

Options

A. 62700
B. 67700
C. 61200
D. 67200

Answer

To solve the problem of arranging the letters of the word "FORECAST" taken 3 at a time and the letters of the word "MILKY" taken 2 at a time, we will follow these steps:

### Step 1: Determine the number of letters in each word
- The word "FORECAST" has 8 letters.
- The word "MILKY" has 5 letters.

### Step 2: Select letters from each word
- We need to select 3 letters from "FORECAST". The number of ways to do this is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of letters and \( r \) is the number of letters to choose.

\[ \text{Ways to choose 3 letters from FORECAST} = \binom{8}{3} \]

- We also need to select 2 letters from "MILKY". The number of ways to do this is:

\[ \text{Ways to choose 2 letters from MILKY} = \binom{5}{2} \]

### Step 3: Calculate the combinations
Using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):

1. Calculate \( \binom{8}{3} \):

\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

2. Calculate \( \binom{5}{2} \):

\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10 \]

### Step 4: Arrange the selected letters
After selecting 3 letters from "FORECAST" and 2 letters from "MILKY", we will have a total of 5 letters. The number of ways to arrange these 5 letters is given by \( 5! \):

\[ 5! = 120 \]

### Step 5: Calculate the total number of arrangements
Now, we can find the total number of arrangements by multiplying the number of ways to choose the letters by the number of arrangements:

\[ \text{Total arrangements} = \binom{8}{3} \times \binom{5}{2} \times 5! = 56 \times 10 \times 120 \]

### Step 6: Perform the final calculation
Calculating the total:

\[ 56 \times 10 = 560 \]
\[ 560 \times 120 = 67200 \]

### Final Answer
Thus, the total number of ways to arrange the letters of the word "FORECAST" taken 3 at a time and the word "MILKY" taken 2 at a time is **67200**.

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