Fundamental Principle of Counting — JEE Main Mathematics MCQs with Solutions
Free JEE Main Mathematics Fundamental Principle of Counting MCQs with step-by-step solutions (16 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
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.
Q2 — 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.
Q3 — 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.
Q4 — 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.
Q5 — 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.
Q6 — 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.
Q7 — 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.
Q8 — 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.
Q9 — 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.
Q10 — 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.
Q11 — 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.
Q12 — 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.
Q13 — 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.
Q14 — 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.
Q15 — 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.
Q16 — 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.