The number of 4 digit numbers that can be made using exactly three digits out of 1,2,3,4,5, and 6.

Options

A. 380
B. 280
C. 240
D. 300

Answer

To solve the problem of finding the number of 4-digit numbers that can be made using exactly three digits from the set {1, 2, 3, 4, 5, 6}, we can follow these steps:

### Step 1: Select 3 Digits from 6
We need to select 3 digits from the 6 available digits. The number of ways to choose 3 digits from 6 can be calculated using the combination formula:

\[ \text{Number of ways to select 3 digits} = \binom{6}{3} \]

Calculating this gives:

\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 2: Form 4-Digit Numbers with Selected Digits
Now, we need to form 4-digit numbers using the 3 selected digits. Since we are using exactly 3 digits, one of these digits must be repeated.

To form a 4-digit number with 3 digits (let's call them A, B, and C), we can have the following arrangements:
- One digit appears twice, and the other two digits appear once each.

### Step 3: Arrangements of the Digits
The total number of arrangements of the digits can be calculated using the formula for permutations of multiset:

\[ \text{Number of arrangements} = \frac{4!}{2! \cdot 1! \cdot 1!} \]

Calculating this gives:

\[ \frac{4!}{2! \cdot 1! \cdot 1!} = \frac{24}{2 \cdot 1 \cdot 1} = 12 \]

### Step 4: Total Combinations
Now, we multiply the number of ways to select the digits by the number of arrangements:

\[ \text{Total 4-digit numbers} = \text{Ways to select 3 digits} \times \text{Arrangements} \]

Substituting the values we calculated:

\[ \text{Total 4-digit numbers} = 20 \times 12 = 240 \]

### Final Answer
Thus, the total number of 4-digit numbers that can be formed using exactly three digits from the set {1, 2, 3, 4, 5, 6} is **240**.

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