Question:
If four digit numbers are formed by using the digits 1, 2, 3, 4, 5, 6, 7 without repetition, then out of these numbers, the numbers exactly divisible by 25 are
If four digit numbers are formed by using the digits 1, 2, 3, 4, 5, 6, 7 without repetition, then out of these numbers, the numbers exactly divisible by 25 are
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Divisibility by 25: Check only the last two digits ($00, 25, 50, 75$).
Updated On: Apr 26, 2026
- 20
- 40
- 50
- 51
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The Correct Option is B
Solution and Explanation
Step 1: Concept
A number is divisible by 25 if its last two digits are divisible by 25. Using digits $\{1, 2, 3, 4, 5, 6, 7\}$, the possible last two digits are $25$ and $75$.
Step 2: Case Analysis
- **Case 1 (Ends in 25):** Last two places are fixed (1 way). Remaining 2 places can be filled from $\{1, 3, 4, 6, 7\}$ in $^5P_2 = 5 \times 4 = 20$ ways. - **Case 2 (Ends in 75):** Last two places are fixed (1 way). Remaining 2 places can be filled from $\{1, 2, 3, 4, 6\}$ in $^5P_2 = 5 \times 4 = 20$ ways.
Step 3: Total
Total numbers $= 20 + 20 = 40$.
Final Answer: (B)
A number is divisible by 25 if its last two digits are divisible by 25. Using digits $\{1, 2, 3, 4, 5, 6, 7\}$, the possible last two digits are $25$ and $75$.
Step 2: Case Analysis
- **Case 1 (Ends in 25):** Last two places are fixed (1 way). Remaining 2 places can be filled from $\{1, 3, 4, 6, 7\}$ in $^5P_2 = 5 \times 4 = 20$ ways. - **Case 2 (Ends in 75):** Last two places are fixed (1 way). Remaining 2 places can be filled from $\{1, 2, 3, 4, 6\}$ in $^5P_2 = 5 \times 4 = 20$ ways.
Step 3: Total
Total numbers $= 20 + 20 = 40$.
Final Answer: (B)
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