Determine the number of 5 digit numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 that are divisible by a) 2 b) 5
Answers
Given : Digits 1, 2, 3, 4, 5, 6 and 5 Digit number Divisible by 2 & 5
To find : Determine the number of 5 digit numbers
Solution:
digits 1, 2, 3, 4, 5, 6
considering repetition of Digit
Divisible by 2 if last Digit is 2 , 4 or 6
hence 3 Ways
1st to 4th Each Digit can be in 6 ways
Hence total possible = 6*6*6*6*3 = 3888
Divisible by 5 if last Digit is 5
hence 1 Way
1st to 4th Each Digit can be in 6 ways
Hence total possible = 6*6*6*6*1 = 1296
if Repetition of digits not allowed
Divisible by 2
6 cases each time 1 Digit not used
1 not used - last digit 3 ways 2 , 4 , 6 and remaining 4 Digits in 4! ways
2 not used - last digit 2 ways 4 , 6 and remaining 4 Digits in 4! ways
3 not used - last digit 3 ways 2 , 4 , 6 and remaining 4 Digits in 4! ways
4 not used - last digit 2 ways 2 , 6 and remaining 4 Digits in 4! ways
5 not used - last digit 3 ways 2 , 4 , 6 and remaining 4 Digits in 4! ways
6 not used - last digit 2 ways 2 , 4 and remaining 4 Digits in 4! ways
= 3 *( 4! * 3 + 4! * 2)
= 3 * ( 120)
= 360
other way last Digits can be selected in 3 ways ( 2 , 4 , 6)
remaining 4 Digits out of 5 Digits in ⁵P₄ = 5! = 120 ways
3 * 120 = 360 numbers
if Repetition of digits not allowed
Divisible by 5
Last Digit = 5 remaining ⁵P₄ = 5! = 120 ways
Hence 120 numbers
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