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)
P(n, r) = \(\frac{n!}{(n-r)!}\)
∴ 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)
P(n, r) = \(\frac{n!}{(n-r)!}\)
∴ 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
