# Mathematics Questions and Answers – Fundamental Principle of Counting

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Fundamental Principle of Counting”.

1. If an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways, then the total numbers of occurrence of the events in the given order is __________________
a) m + n
b) m – n
c) m*n
d) m/n

Explanation: By the fundamental principle of counting, if an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways, then the total numbers of occurrence of the events in the given order is m*n.

2. A child has 2 pencil and 3 erasers. In how many ways he can take a pencil and an eraser?
a) 5
b) 6
c) 8
d) 9

Explanation: By the fundamental principle of counting, if an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways, then the total numbers of occurrence of the events in the given order is m*n.
So, if pencil can be taken in 2 ways and eraser can be taken in 3 ways then number of ways in which he can take a pencil and eraser are 2*3 = 6.

3. If there are 4 paths to travel from Delhi to Kanpur, then in how many ways a person can travel from Delhi to Kanpur and came back to Delhi?
a) 4
b) 8
c) 12
d) 16

Explanation: If there are 4 paths to travel from Delhi to Kanpur then number of paths to travel from Kanpur to Delhi are 4. So, by the fundamental principle of counting, total number of ways to go and come back are 4*4 = 16.
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4. If there are 4 paths to travel from Delhi to Kanpur, then in how many ways a person can travel from Delhi to Kanpur and came back to Delhi via same path?
a) 4
b) 8
c) 12
d) 16

Explanation: If there are 4 paths to travel from Delhi to Kanpur and person take one path then in coming back journey only one option is there. So, by the fundamental principle of counting, total number of ways a person can travel from Delhi to Kanpur and came back to Delhi via same path are 4*1 = 4.

5. If there are 4 paths to travel from Delhi to Kanpur, then in how many ways a person can travel from Delhi to Kanpur and came back to Delhi via different path?
a) 4
b) 8
c) 12
d) 16

Explanation: If there are 4 paths to travel from Delhi to Kanpur and person take one path then in coming back journey only 3 paths are remaining. So, by the fundamental principle of counting, total number of ways a person can travel from Delhi to Kanpur and came back to Delhi via different path are 4 * 3 = 12.

6. Find the number of 5 letter words which can be formed from word PULSE without repetition.
a) 20
b) 60
c) 120
d) 240

Explanation: PULSE is a 5 letters word. By the fundamental principle of counting, total number of possible words without repetition are 5*4*3*2*1 = 120. 7. Find the number of 5 letter words which can be formed from word PULSE if repetition is allowed.
a) 25
b) 120
c) 125
d) 3125

Explanation: PULSE is a 5 letters word. By the fundamental principle of counting, total number of possible words with repetition allowed are 5*5*5*5*5 = 3125. 8. Find the number of 4 letter words which can be formed from word PULSE without repetition.
a) 20
b) 60
c) 120
d) 240

Explanation: PULSE is a 5 letters word. By the fundamental principle of counting, total number of possible words without repetition are 5*4*3*2 = 120. 9. Find the number of 4 letter words which can be formed from word PULSE if repetition is allowed.
a) 120
b) 125
c) 625
d) 3125

Explanation: PULSE is a 5 letters word. By the fundamental principle of counting, total number of possible words with repetition allowed are 5*5*5*5 = 625. 10. How many 5-digit numbers are possible without repetition of digits?
a) 27216
b) 50400
c) 100000
d) 90000

Explanation: Total digits are 0-9 i.e.10. In a 5-digit number 0 cannot be at first place.
So, by the fundamental principle of counting, total numbers possible are 9*9*8*7*6 = 27216. 11. How many 5-digit numbers are possible if repetition of digits is allowed?
a) 27216
b) 50400
c) 100000
d) 90000

Explanation: Total digits are 0-9 i.e.10. In a 5-digit number 0 cannot be at first place.
So, by the fundamental principle of counting, total numbers possible are 9*10*10*10*10=90000. 12. A passcode is made of 5 digits. How many maximum numbers of ways incorrect passcode is entered?
a) 27215
b) 50399
c) 99999
d) 89999

Explanation: Total digits are 0-9 i.e.10. In passcode, 0 may also occur at first place.
All places are filled in 10 ways.
So, by the fundamental principle of counting, total numbers possible are 10*10*10*10*10=100000.
Maximum number of incorrect pass code entered = 100000-1 = 99999.

13. How many 4 digits even numbers are possible from digits 1 to 9 if repetition is not allowed?
a) 6561
b) 2016
c) 1344
d) 2916

Explanation: Total digits are 1-9 i.e.9. Even number means 2,4,6,8 at unit’s place.
So, 4 possible digits for unit place, after a digit is placed at unit place 8 digits are remaining.
8 ways to fill 1st position, 7 ways to fill 2nd position, 6 ways to fill th3 3rd position.
Hence, by the fundamental principle of counting, total possible numbers are 8*7*6*4 = 1344. 14. How many 4 digits even numbers are possible from digits 1 to 9 if repetition is allowed?
a) 6561
b) 2016
c) 1344
d) 2916

Explanation: Total digits are 1-9 i.e.9. Even number means 2,4,6,8 at unit’s place.
So, 4 possible digits for unit place. Rest all positions are filled in 9 ways.
Hence, by the fundamental principle of counting, total possible numbers are 9*9*9*4 = 2916. 15. If a code involves one letter of English alphabet at first place and a number at second place. Numbers can be used from 1 to 9. How many codes are possible using the letters of alphabet and numbers?
a) 234
b) 260
c) 314
d) 324

Explanation: Since there are 26 letters in English alphabet and 9 possible digits from 1 to 9.
So, by the fundamental principle of counting, total possible codes are 26*9 = 234.

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