The number of 4 digit numbers that can be made using exactly two digits out of 1,2,3,4,5,6 and 7

Options

A. 126
B. 210
C. 284
D. 294

Explanation

To find the number of 4-digit numbers that can be made using exactly two digits from the set {1, 2, 3, 4, 5, 6, 7}, we can follow these steps:

### Step 1: Choose 2 Digits
We need to select exactly 2 digits from the available 7 digits. The number of ways to choose 2 digits from 7 can be calculated using the combination formula:

\[ \text{Number of ways to choose 2 digits} = \binom{7}{2} \]

Calculating this gives:

\[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

### Step 2: Form 4-Digit Numbers with the Chosen Digits
Once we have chosen 2 digits, we need to form 4-digit numbers using these digits. Each digit can be either of the two chosen digits. Therefore, for each of the 4 positions in the number, we have 2 choices (either of the two digits).

The total number of combinations for the 4-digit number is:

\[ 2^4 = 16 \]

### Step 3: Exclude Invalid Cases
However, we need to exclude the cases where all four digits are the same. There are exactly 2 such cases (one for each digit). Therefore, we subtract these 2 cases from the total combinations:

\[ \text{Valid combinations} = 16 - 2 = 14 \]

### Step 4: Calculate the Total Number of 4-Digit Numbers
Now, we multiply the number of ways to choose the digits by the number of valid combinations of those digits:

\[ \text{Total 4-digit numbers} = 21 \times 14 = 294 \]

### Final Answer
Thus, the total number of 4-digit numbers that can be formed using exactly two digits from the set {1, 2, 3, 4, 5, 6, 7} is **294**.

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