Practice Questions
Concept Explanation
The fundamental counting principle states that if one event can occur in m ways and a second independent event in n ways, both together can occur in m × n ways.
Key Rules
- Multiplication rule (AND): total = m × n × …
- Addition rule (OR, mutually exclusive): total = m + n
- Applies when choices are independent and sequential
Example
4 shirts and 5 trousers give 4 × 5 = 20 outfits.
Common Trap
Use addition only when choices are mutually exclusive (either/or), not when both are required.
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Q1 EasyA person has 4 different shirts and 5 different trousers. If an outfit consists of one shirt and one trouser, how many different outfits can be formed?ExplanationBy the multiplication rule, each shirt can be paired with any of the 5 trousers. Total outfits = 4 × 5 = 20. Hence the answer is 20.
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Q2 EasyFrom city X to city Y there are 3 roads, and from city Y to city Z there are 4 roads. In how many different ways can a person travel from X to Z via Y?ExplanationEach route from X to Y can be combined with each route from Y to Z. By the fundamental counting principle, total ways = 3 × 4 = 12. Hence the answer is 12.
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Q3 EasyA true/false quiz has 6 questions and each question has exactly two possible answers. How many different answer sheets are possible?ExplanationEach question can be answered in 2 ways, so total answer sheets = 2^6 = 64. Hence the answer is 64.
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Q4 EasyEither a mathematics book (4 titles) or a science book (6 titles) may be chosen for reading. How many choices does a student have?ExplanationThe two categories are mutually exclusive (either/or). By the addition rule, total choices = 4 + 6 = 10. Hence the answer is 10.
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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?ExplanationA number is divisible by 5 if its units digit is 0 or 5. From the given digits, only 5 can be in the units place (1 way). The hundreds and tens places are then filled from the remaining 5 digits with no repetition: 5 × 4 = 20. By the fundamental counting principle, total = 1 × 5 × 4 = 20. Hence the answer is 20.
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Q6 MediumIn how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?ExplanationMen can be chosen in 7C5 ways and women in 3C2 ways. Using 7C5 = 21 and 3C2 = 3, total groups = 21 × 3 = 63 by the multiplication rule. Hence the answer is 63.
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Q7 MediumA meal consists of one appetizer chosen from 6 options, one main course from 8 options, and one dessert from 4 options. How many different meals are possible?ExplanationThe three choices are independent. Total meals = 6 × 8 × 4 = 192. Hence the answer is 192.
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Q8 MediumA vehicle is available in 3 colours, 5 models and 2 engine types. How many different variants can be produced?ExplanationEach colour–model–engine combination is a variant. Total = 3 × 5 × 2 = 30. Hence the answer is 30.
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Q9 MediumFrom A to B there are 2 roads, from B to C there are 3 roads, and from C to D there are 2 roads. In how many ways can a person travel from A to D via B and C?ExplanationApply the multiplication rule to the three successive segments: 2 × 3 × 2 = 12 ways. Hence the answer is 12.
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Q10 HardHow many signals can be made with 5 different flags when any number of them can be used at a time?ExplanationAny number of flags means using 1, 2, 3, 4 or all 5 flags, and order matters for a signal. Number of signals = 5P1 + 5P2 + 5P3 + 5P4 + 5P5 = 5 + (5 × 4) + (5 × 4 × 3) + (5 × 4 × 3 × 2) + 5! = 5 + 20 + 60 + 120 + 120 = 325. Hence the answer is 325.
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Q11 HardIn a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?ExplanationTotal ways to select 4 children from 10 = 10C4 = 210. Ways with no boy (all 4 girls) = 4C4 = 1. Required = 210 − 1 = 209. Hence the answer is 209.
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Q12 HardA password has 2 letters (A–Z) followed by 2 digits (0–9), and repetition is allowed. How many such passwords can be formed?ExplanationEach letter place has 26 choices and each digit place has 10 choices. Total = 26 × 26 × 10 × 10 = 676 × 100 = 67600. Hence the answer is 67600.
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Q13 HardFrom a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there in the committee. In how many ways can it be done?ExplanationCases: (3 men, 2 women) + (4 men, 1 woman) + (5 men, 0 women). Compute 7C3 × 6C2 = 35 × 15 = 525; 7C4 × 6C1 = 35 × 6 = 210; 7C5 × 6C0 = 21 × 1 = 21. Total = 525 + 210 + 21 = 756. Hence the answer is 756.
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Q14 HardA committee of 5 members is to be formed out of 6 men and 4 women. In how many ways can this be done if at least 2 women must be included?ExplanationCases with at least 2 women: (3 men, 2 women) + (2 men, 3 women) + (1 man, 4 women). Compute 6C3 × 4C2 = 20 × 6 = 120; 6C2 × 4C3 = 15 × 4 = 60; 6C1 × 4C4 = 6 × 1 = 6. Total = 120 + 60 + 6 = 186. Hence the answer is 186.
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DS1 Easy
How many different outfits can be formed if each outfit uses one shirt and one trouser?
Statement IThere are 4 different shirts and 5 different trousers available.
Statement IIThe number of trousers equals the number of shirts.
ExplanationThe question already requires one shirt and one trouser per outfit. Statement I alone gives outfits = 4 × 5 = 20. Statement II alone does not give the actual counts. Answer: A. -
DS2 Easy
How many ways can a traveller go from P to R via Q?
Statement IFrom P to Q there are 3 distinct routes.
Statement IIFrom Q to R there are 5 distinct routes.
ExplanationStatement I alone does not give routes from Q to R. Statement II alone does not give routes from P to Q. Together, by the fundamental counting principle, ways = 3 × 5 = 15. Both are needed. Answer: C.
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DS3 Easy
How many answer sheets are possible on a true/false quiz. What is that number?
Statement IThe quiz consists of exactly 8 questions.
Statement IIEach question has exactly two possible answers (true or false).
ExplanationStatement I alone does not specify how many choices each question has. Statement II alone does not give the number of questions. Together, answer sheets = 2^8 = 256. Both are needed. Answer: C. -
DS4 Medium
How many different meals can be ordered from a fixed menu structure. What is that number?
Statement IThere are 5 appetizers and 6 main courses on the menu.
Statement IIEach meal must include one appetizer, one main course, and exactly one of 3 desserts.
ExplanationStatement I alone lacks the dessert count and the meal structure for desserts. Statement II alone already includes appetizers (5), mains (6) and desserts (3), so meals = 5 × 6 × 3 = 90. Answer: B.