The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is:
Show Hint
- 5719
- 4608
- 5720
- 4607
The Correct Option is B
Solution and Explanation
The first digit must be \( \geq 5 \) to ensure the number is greater than 50000. This restricts the first digit to 5, 6, or 7.
For each valid first digit \( d_1 \) (5, 6, or 7), determine possible last digits \( d_5 \) such that their sum \( d_1 + d_5 \leq 8 \):
\[ \begin{aligned} \text{For } d_1 = 5: & \quad \text{Possible } d_5 \text{ are } 0, 1, 2, 3 \quad \text{(4 choices)} \\ \text{For } d_1 = 6: & \quad \text{Possible } d_5 \text{ are } 0, 1, 2 \quad \text{(3 choices)} \\ \text{For } d_1 = 7: & \quad \text{Possible } d_5 \text{ are } 0, 1 \quad \text{(2 choices)} \end{aligned} \]Each of the middle three digits (\(d_2, d_3, d_4\)) can be any of the 8 digits (0-7). Calculating the combinations for each case:
\[ \begin{aligned} \text{For } d_1 = 5: & \quad 4 \times 8^3 = 2048 \\ \text{For } d_1 = 6: & \quad 3 \times 8^3 = 1536 \\ \text{For } d_1 = 7: & \quad 2 \times 8^3 = 1024 \end{aligned} \]The total number of such 5-digit numbers greater than 50000, formed under the given constraints, is 4608.
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