Q. 305.0( 2 Votes )

How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 5, 6, 7 if no digits is repeated?

Answer :

Given: Digits which can be used to make numbers are 1, 2, 3, 4, 5, 6 and 7


The number of these digits are 7


To find: total number of three-digit even numbers with no digit repeated


Even numbers are those numbers whose unit’s place is even


fix the position of 1 even number at unit’s place at one time


Even numbers are: 2, 4 and 6


Case 1:


Fix position of 2 at unit’s place



Remaining numbers = 6


Arrange these 6 numbers at remaining 2 places


Formula used:


Number of arrangements of n things taken r at a time = P(n, r)



The total numbers ending with 2 are =


= the number of arrangements of 6 things taken 2 at a time


= P(6, 2)





= 6 × 5


= 30


Case 2:


Fix position of 4 at unit’s place



Remaining numbers = 6


Now, arrange these 6 numbers in remaining 2 places


Formula used:


Number of arrangements of n things taken r at a time = P(n, r)



The total numbers ending with 2 are =


= the number of arrangements of 6 things taken 2 at a time


= P(6, 2)





= 6 × 5


= 30


Similarly,


When you fix position of 6 at unit’s place, 30 more numbers will be formed.


Hence, total number of three-digit even numbers with no digit repeated are, 30 + 30 + 30 = 90


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