Question

How many 3-digit numbers can be formed from the digits 2, 3, 5, 8 and 9, without repetition, which are exactly divisible by 4? ​

Answer 1

Step-by-step explanation:

Given that:

How many 3-digit numbers can be formed from the digits 2, 3, 5, 8 and 9, without repetition, which are exactly divisible by 4?

To find:No of ways to arrange so that numbers are exactly divisible by 4

Solution:

From the given numbers 2,3,5,8 and 9

2 and 8 can occupy last (unit digits)

Reason: A number is divisible by 4 only if it's last two digits are divisible by 4.

Case 1: Fix 2 at unit place.

Now we have four numbers out of which we have to choose 2 digits for tenth and hundredth place

n= 4(3,5,8,9)

r= 2(only 2 place vacant)

No. of methods:

$^{4} P_2 = \frac{4!}{(4 - 2)!} \\ \\ = \frac{4!}{2!} \\ \\ = \frac{4\:\times\:3\times\:2!}{2!} \\ \\=12\:methods$

Out of which the numbers which have 8 at tenth place are not divisible by 4

(382,582,982)= 3 numbers should remove from 12 ways

Total ways to arrange = 12-3= 9 ways

Case2: Fix 8 at unit place.

Now we have four numbers out of which we have to choose 2 for tenth place

because 38,58,98 are not divisible by 4

n= 3(3,5,9)

r= 1(only 1 place vacant)

No. of methods:

$^{3} P_1 = \frac{3!}{(3 - 1)!} \\ \\ = \frac{3!}{2!} \\ \\ = 3\:ways$

Total possible ways:

No. of ways from case1+ No. of ways from case2

=9+3

= 12

Total 3 digits numbers formed from given 5 digits are 12 numbers, which are divisible by 4.

Hope it helps you.

Answer 2

Given :   3-digit numbers can be formed from the digits 2, 3, 5, 8 and 9, without repetition

To find : How many 3-digit numbers can be formed

Solution:

Number divisible by 4 if

last two two digits divisible by 4  

Last digit can be 2 or   8  

if last digit is 2

32 - Divisible  by 4

52 , Divisible by 4

82 - Not Divisible by 4

92 - Divisible by 4  

if last digit is 8

28  divisible  by 4

38 , 58 & 98 are not divisible by 4

28  , 32  , 52  ,  92   are divisible by 4  

in each case 1st digit can be in 3 ways  

so 3 * 4 = 12  numbers

328  , 528  , 928    divisible by 4  

532  , 832  ,  932     divisible by 4  

352 , 852  ,   952    divisible by 4  

392  , 592 , 892      divisible by 4  

12  , 3-digit numbers can be formed from the digits 2, 3, 5, 8 and 9, without repetition, which are exactly divisible by 4

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