Practice Questions
Concept Explanation
Number formation uses digits with rules on repetition, leading zeros, and divisibility. Count valid digit arrangements carefully.
Key Strategies
- No leading zero: first digit from 1–9, rest from 0–9
- Without repetition: use nPr on digit positions
- With repetition: use nr when allowed
Example
2-digit numbers from {1,2,3,4,5} without repetition: 5 × 4 = 20.
Common Trap
Exclude numbers with leading zero when forming multi-digit numbers.
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Q1 EasyHow many 2-digit numbers can be formed from the digits 1, 2, 3, 4, 5 if no digit is repeated?ExplanationTens place: 5 choices; units place: 4 remaining choices. Total = 5 × 4 = 20. Hence the answer is 20.
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Q2 EasyHow many 3-digit numbers can be formed from {1,2,3,4,5} if repetition is allowed?ExplanationEach of the 3 places has 5 choices. Total = 5³ = 125. Hence the answer is 125.
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Q3 EasyHow many 4-digit numbers can be formed from {1,2,3,4,5,6} without repetition?Explanation4 positions from 6 distinct digits: 6P4 = 6 × 5 × 4 × 3 = 360. Hence the answer is 360.
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Q4 EasyHow many 5-digit numbers can be formed from {1,2,3,4,5} without repetition?ExplanationAll 5 distinct digits used: 5! = 120. Hence the answer is 120.
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Q5 MediumHow many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?ExplanationFor divisibility by 5 with this digit set, the units digit must be 5. Fix units as 5 (1 way). Hundreds place: 5 choices from {2,3,6,7,9}; tens place: 4 remaining choices. Total = 1 × 5 × 4 = 20. Hence the answer is 20.
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Q6 MediumHow many 3-digit numbers can be formed from {0,1,2,3,4} without repetition and without a leading zero?ExplanationHundreds digit: 4 choices (1–4). Then tens: 4 remaining (including 0), units: 3 remaining. Total = 4 × 4 × 3 = 48. Hence the answer is 48.
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Q7 MediumHow many 4-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if repetition of digits is not allowed?ExplanationThousands digit cannot be 0: 9 choices (1–9). Hundreds: 9 remaining (including 0 but excluding the used digit). Tens: 8; units: 7. Total = 9 × 9 × 8 × 7 = 4536. Hence the answer is 4536.
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Q8 MediumHow many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7 if no digit is repeated?ExplanationUnits digit must be even: 2, 4, or 6 → 3 choices. Hundreds: 5 remaining digits; tens: 4 remaining. Total = 3 × 5 × 4 = 60. Hence the answer is 60.
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Q9 MediumHow many 2-digit numbers from {0,1,2,3} without repetition (no leading zero) are possible?ExplanationTens digit: 3 choices (1,2,3). Units: 3 remaining digits (including 0). Total = 3 × 3 = 9. Hence the answer is 9.
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Q10 MediumHow many 3-digit numbers from {1,2,3,4,5} without repetition are divisible by 5?ExplanationUnits must be 5. Hundreds: 4 choices; tens: 3 remaining. Total = 4 × 3 = 12. Hence the answer is 12.
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Q11 HardHow many numbers greater than 3000 can be formed with the digits 1, 2, 3, 4, 5 if repetition of digits is not allowed?Explanation4-digit numbers greater than 3000: thousands digit ∈ {3,4,5} (3 choices); remaining 3 places: 4P3 = 4 × 3 × 2 = 24. So 3 × 24 = 72. 5-digit numbers using all digits: 5! = 120 (all exceed 3000). Total = 72 + 120 = 192. Hence the answer is 192.
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Q12 HardHow many 4-digit numbers can be formed with the digits 0, 1, 2, 3, 4, 5 if repetition of digits is not allowed?ExplanationThousands digit: 5 choices (1–5). Hundreds: 5 remaining (including 0); tens: 4; units: 3. Total = 5 × 5 × 4 × 3 = 300. Hence the answer is 300.
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Q13 HardHow many 4-digit even numbers can be formed from {1,2,3,4,5} without repetition?ExplanationUnits even: 2 or 4 → 2 choices. Remaining three places: 4P3 = 24. Total = 2 × 24 = 48. Hence the answer is 48.
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Q14 HardHow many 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5 if repetition of digits is not allowed?ExplanationHundreds digit cannot be 0: 5 choices (1–5). Tens: 5 remaining digits (including 0); units: 4 remaining. Total = 5 × 5 × 4 = 100. Hence the answer is 100.
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DS1 Easy
In a campus number-formation drill, how many 2-digit numbers can be formed from {1,2,3,4,5} without repetition?
Statement IDigits cannot be repeated.
Statement IIThe set of available digits is {1,2,3,4,5}.
ExplanationNeed both the set and the no-repetition rule: 5 × 4 = 20. Answer: C. -
DS2 Easy
How many 4-digit numbers without repetition use digits 0–9.
Statement IThe thousands digit cannot be 0.
Statement IIDigits may not be repeated.
ExplanationTogether: 9 × 9 × 8 × 7 = 4536. Answer: C.
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DS3 Medium
How many 3-digit numbers without a leading zero can be formed?
Statement IDigits are chosen from {0,1,2,3,4}.
Statement IINo digit is repeated.
ExplanationTogether: hundreds has 4 choices, then 4 and 3 → 48. Answer: C. -
DS4 Medium
An HR quiz asks how many 3-digit numbers divisible by 5 can be formed with no digit repeated.
Statement IThe available digits are 2, 3, 5, 6, 7 and 9.
Statement IIOnly the units digit is known to be 5.
ExplanationStatement I alone with the stem conditions: units must be 5, then 5 × 4 = 20. Statement II alone lacks the full digit set for the other places. Answer: A.