Fundamental Principle of Counting — JEE Main Mathematics MCQs with Solutions
Free JEE Main Mathematics Fundamental Principle of Counting MCQs with step-by-step solutions (30 questions). Part of Permutation and Combination. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Fundamental Principle of Counting · easy · numerical
There are 4 roads from town A to town B and 3 roads from town B to town C. In how many ways can a person travel from A to C via B?
A. 16
B. 24
C. 12 ✓ Correct
D. 7
Solution: By the multiplication principle, the journey A to B can be done in 4 ways and B to C in 3 ways, giving 4 × 3 = 12 ways.
Q2 — Fundamental Principle of Counting · easy · numerical
A boy has 5 different shirts and 4 different trousers. In how many ways can he choose one shirt and one trouser to wear?
A. 16
B. 25
C. 20 ✓ Correct
D. 9
Solution: A shirt can be chosen in 5 ways and a trouser in 4 ways, so the outfit can be chosen in 5 × 4 = 20 ways.
Q3 — Fundamental Principle of Counting · easy · numerical
How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 if no digit is repeated?
A. 840 ✓ Correct
B. 2401
C. 5040
D. 210
Solution: The four places can be filled in 7, 6, 5 and 4 ways respectively, giving 7 × 6 × 5 × 4 = 840 numbers.
Q4 — Fundamental Principle of Counting · easy · numerical
How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 if repetition of digits is allowed?
A. 108
B. 120
C. 216 ✓ Correct
D. 18
Solution: With repetition allowed, each of the three places can be filled in 6 ways, giving 6 × 6 × 6 = 216 numbers.
Q5 — Fundamental Principle of Counting · easy · numerical
How many even 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5 if no digit is repeated?
A. 60
B. 48
C. 12
D. 24 ✓ Correct
Solution: The units place must be 2 or 4 (2 ways); the remaining two places can then be filled in 4 × 3 = 12 ways, giving 2 × 12 = 24 numbers.
Q6 — Fundamental Principle of Counting · easy · numerical
A signal is made by hoisting 2 flags, one above the other, chosen from 5 flags of different colours. How many different signals are possible?
A. 9
B. 25
C. 10
D. 20 ✓ Correct
Solution: The upper flag can be chosen in 5 ways and the lower flag in 4 ways, so there are 5 × 4 = 20 signals.
Q7 — Fundamental Principle of Counting · easy · numerical
A coin is tossed 4 times and the sequence of heads and tails is recorded. How many different outcomes are possible?
A. 16 ✓ Correct
B. 8
C. 4
D. 24
Solution: Each toss has 2 possible results, so the number of outcomes is 2 × 2 × 2 × 2 = 2⁴ = 16.
Q8 — Fundamental Principle of Counting · easy · numerical
Two dice, one red and one blue, are rolled together. How many different outcomes are possible?
A. 12
B. 30
C. 21
D. 36 ✓ Correct
Solution: Each die can land in 6 ways, so the pair of dice gives 6 × 6 = 36 outcomes.
Q9 — Fundamental Principle of Counting · easy · numerical
How many 3-letter codes can be formed from the 26 letters of the English alphabet if letters may be repeated?
A. 17576 ✓ Correct
B. 15600
C. 676
D. 78
Solution: Each of the 3 positions can be filled by any of the 26 letters, so the count is 26 × 26 × 26 = 17576.
Q10 — Fundamental Principle of Counting · easy · numerical
How many 4-digit ATM PINs are possible if each digit can be any digit from 0 to 9 and digits may repeat?
A. 6561
B. 10000 ✓ Correct
C. 9000
D. 5040
Solution: Each of the 4 positions has 10 choices, so the number of PINs is 10 × 10 × 10 × 10 = 10000.
Q11 — Fundamental Principle of Counting · easy · numerical
In how many ways can 4 different letters be dropped into 5 mailboxes, if each letter can go into any mailbox?
A. 120
B. 625 ✓ Correct
C. 1024
D. 20
Solution: Each of the 4 letters can be placed in any of the 5 mailboxes, so the count is 5 × 5 × 5 × 5 = 5⁴ = 625.
Q12 — Fundamental Principle of Counting · easy · numerical
A test has 5 multiple-choice questions, each with 4 options. In how many different ways can a student answer all 5 questions?
A. 1024 ✓ Correct
B. 625
C. 120
D. 20
Solution: Each question can be answered in 4 ways, so all five can be answered in 4 × 4 × 4 × 4 × 4 = 4⁵ = 1024 ways.
Q13 — Fundamental Principle of Counting · medium · numerical
There are 5 different roads between two villages. In how many ways can a man go from one village to the other and return by a different road?
A. 9
B. 20 ✓ Correct
C. 10
D. 25
Solution: He can go by any of the 5 roads and must return by one of the remaining 4 roads, giving 5 × 4 = 20 ways.
Q14 — Fundamental Principle of Counting · medium · numerical
There are 3 routes from city P to city Q and 4 routes from city Q to city R. In how many ways can a person travel from P to R via Q and come back to P via Q, without using any route twice?
A. 24
B. 84
C. 144
D. 72 ✓ Correct
Solution: Going takes 3 × 4 = 12 ways; returning, the Q-to-P route must differ (2 left) and the R-to-Q route must differ (3 left), giving 12 × 3 × 2 = 72 ways.
Q15 — Fundamental Principle of Counting · medium · numerical
How many 4-digit numbers with all distinct digits can be formed using the digits 1 to 9 (zero not allowed)?
A. 4536
B. 6561
C. 3024 ✓ Correct
D. 2520
Solution: The four places can be filled in 9, 8, 7 and 6 ways respectively, giving 9 × 8 × 7 × 6 = 3024 numbers.
Q16 — Fundamental Principle of Counting · medium · numerical
How many even 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 without repetition?
A. 840
B. 360 ✓ Correct
C. 240
D. 720
Solution: The units digit must be 2, 4 or 6 (3 ways); the remaining three places are filled from the remaining 6 digits in 6 × 5 × 4 = 120 ways, giving 3 × 120 = 360.
Q17 — Fundamental Principle of Counting · medium · numerical
How many 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5 without repetition?
A. 125
B. 120
C. 100 ✓ Correct
D. 60
Solution: The hundreds place cannot be 0, so it has 5 choices; the tens place then has 5 remaining choices and the units place 4, giving 5 × 5 × 4 = 100 numbers.
Q18 — Fundamental Principle of Counting · medium · numerical
How many odd natural numbers less than 1000 can be formed using the digits 1, 2, 3, 4, 5, with repetition of digits allowed?
A. 75
B. 105
C. 93 ✓ Correct
D. 90
Solution: The units digit must be 1, 3 or 5. One-digit numbers: 3; two-digit: 5 × 3 = 15; three-digit: 5 × 5 × 3 = 75. By the addition principle, total = 3 + 15 + 75 = 93.
Q19 — Fundamental Principle of Counting · medium · numerical
How many different signals can be generated by arranging 2, 3 or 4 flags vertically on a pole, chosen from 5 flags of different colours?
A. 240
B. 120
C. 200 ✓ Correct
D. 180
Solution: Signals with 2 flags: 5 × 4 = 20; with 3 flags: 5 × 4 × 3 = 60; with 4 flags: 5 × 4 × 3 × 2 = 120. Total = 20 + 60 + 120 = 200.
Q20 — Fundamental Principle of Counting · medium · numerical
A password has 4 characters. The first character must be one of the 26 letters, and each of the remaining 3 characters can be a letter or a digit (36 choices each), with repetition allowed. How many passwords are possible?
A. 1213056 ✓ Correct
B. 456976
C. 1021020
D. 1679616
Solution: The first character has 26 choices and each of the other three has 36, giving 26 × 36 × 36 × 36 = 26 × 46656 = 1213056 passwords.
Q21 — Fundamental Principle of Counting · medium · numerical
How many 4-digit numbers (from 1000 to 9999) have at least one repeated digit?
A. 3024
B. 5040
C. 4464 ✓ Correct
D. 4536
Solution: Total 4-digit numbers = 9000; those with all distinct digits = 9 × 9 × 8 × 7 = 4536. By complement, at least one repeat = 9000 − 4536 = 4464.
Q22 — Fundamental Principle of Counting · medium · numerical
Three dice of different colours are rolled together. In how many outcomes do at least two of the dice show the same number?
A. 106
B. 120
C. 96 ✓ Correct
D. 216
Solution: Total outcomes = 6 × 6 × 6 = 216; outcomes with all three numbers different = 6 × 5 × 4 = 120. By complement, at least two alike = 216 − 120 = 96.
Q23 — Fundamental Principle of Counting · medium · numerical
A 7-digit telephone number must begin with 67, and no digit may appear more than once in the number. How many such telephone numbers are possible?
A. 32768
B. 6720 ✓ Correct
C. 30240
D. 100000
Solution: The digits 6 and 7 are already used, so the remaining 5 positions are filled from the other 8 digits in 8 × 7 × 6 × 5 × 4 = 6720 ways.
Q24 — Fundamental Principle of Counting · medium · numerical
How many 3-digit numbers divisible by 5 can be formed using the digits 0, 1, 2, 3, 4, 5 without repetition?
A. 40
B. 24
C. 20
D. 36 ✓ Correct
Solution: If units = 0: 5 × 4 = 20 numbers; if units = 5: hundreds has 4 choices (not 0 or 5) and tens has 4, giving 16. Total = 20 + 16 = 36.
Q25 — Fundamental Principle of Counting · medium · numerical
How many 4-letter codes can be formed using the letters of the word MOBILE, without repetition, if the code must end in a vowel?
A. 180 ✓ Correct
B. 360
C. 120
D. 90
Solution: The last place must be O, I or E (3 ways); the first three places are then filled from the remaining 5 letters in 5 × 4 × 3 = 60 ways, giving 3 × 60 = 180 codes.
Q26 — Fundamental Principle of Counting · medium · numerical
In how many ways can 4 people be seated on 6 chairs placed in a row?
A. 360 ✓ Correct
B. 720
C. 24
D. 1296
Solution: The first person can choose a chair in 6 ways, the next in 5, then 4, then 3, giving 6 × 5 × 4 × 3 = 360 ways.
Q27 — Fundamental Principle of Counting · medium · numerical
How many even numbers between 2000 and 3000 can be formed using the digits 1, 2, 3, 4, 5, 6, with no digit repeated?
A. 36
B. 24 ✓ Correct
C. 60
D. 48
Solution: The thousands digit must be 2; the units digit must then be 4 or 6 (2 ways); the middle two places are filled from the remaining 4 digits in 4 × 3 = 12 ways, giving 2 × 12 = 24 numbers.
Q28 — Fundamental Principle of Counting · medium · numerical
In how many ways can 3 different prizes be given to 5 boys, if any boy may receive any number of prizes?
A. 15
B. 125 ✓ Correct
C. 243
D. 60
Solution: Each prize can go to any of the 5 boys, so the prizes can be distributed in 5 × 5 × 5 = 125 ways.
Q29 — Fundamental Principle of Counting · medium · numerical
A coin is tossed 6 times. In how many of the possible outcomes does at least one head appear?
A. 62
B. 64
C. 63 ✓ Correct
D. 32
Solution: Total outcomes = 2⁶ = 64; only one outcome (all tails) has no head. By complement, at least one head occurs in 64 − 1 = 63 outcomes.
Q30 — Fundamental Principle of Counting · medium · numerical
A licence plate consists of 2 letters followed by 3 digits. If the two letters must be different but digits may repeat, how many licence plates are possible?
A. 676000
B. 486720
C. 468000
D. 650000 ✓ Correct
Solution: The letters can be chosen in 26 × 25 = 650 ways and the digits in 10 × 10 × 10 = 1000 ways, giving 650 × 1000 = 650000 plates.