## Dhaka Higher Secondary 2010/9

Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
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BdMO
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### Dhaka Higher Secondary 2010/9

Out of the digits \$1, 2, 3, 4, 5\$, three are chosen to form numbers so that their digits are either in increasing or decreasing order. What is the total number of numbers formed? If one number is chosen from each category so that the sum of those two numbers is maximum, what is that sum?

leonardo shawon
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Location: Dhaka

### Re: Dhaka Higher Secondary 2010/9

decreasing order||| 543 > 432 > 321
increasing order || 123 < 234 < 345

the sum is 888
Ibtehaz Shawon
BRAC University.

long way to go .....

Moon
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### Re: Dhaka Higher Secondary 2010/9

The second part is correct. However, you need to show some logic. Actually we are trying maximize the number. So apparently the largest number \$543\$ works.

Hint for the first part:
what if we forget that the digits can be increasing or decreasing?
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

learn how to write equations, and don't forget to read Forum Guide and Rules.

leonardo shawon
Posts: 169
Joined: Sat Jan 01, 2011 4:59 pm
Location: Dhaka

### Re: Dhaka Higher Secondary 2010/9

1,2,3,4,5.. Among this numbers , we can form 60 (5P3) three digits numbers... If we put 5 on left, the rest 2 place can be filled by 4P2 > 12 ways and numbers are 512, 513, 514, 521, 523, 524, 531, 532, 534, 541, 542, 543. And among this 12 numbers, value of 543 is the largest.

Having 1 on left, rest 4 digits can form 12 , 3 digits number. Among them 123 is the lowest.

## what do u think? Did i give some logic?
Ibtehaz Shawon
BRAC University.

long way to go .....

Moon
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### Re: Dhaka Higher Secondary 2010/9

Still no solution for the first part!
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

learn how to write equations, and don't forget to read Forum Guide and Rules.

Shifat
Posts: 53
Joined: Sun Jul 31, 2011 12:21 pm

### Re: Dhaka Higher Secondary 2010/9

I think I have a solution, the total number of increasing numbers is and decreasing is 20. Here it is:-
While we are writing them as decreasing numbers, then either the first one should be 5,4,3... Now there is three cases
First Value five:- the second value can be 4,3 or 2 .if the second value is 4 then the third value(3,2,1) can be input in 3P1=3 ways, similarly if the second value is three then the third value can be input in 2P1=2 ways, if the second value is 2 then the third value can be input in only 1 way.. so when the first value is five, the number of decreasing numbers is 3+2+1=6
First value four:- By doing the same we can find the number of decreasing numbers is 1+2=3
first value 3:- we can do the same and the result is 1..
So the total number of decreasing numbers is 10... do the same for the increasing numbers.. the sum is 20.

Ragib Farhat Hasan
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Joined: Sun Mar 30, 2014 10:40 pm

### Re: Dhaka Higher Secondary 2010/9

Total no. of numbers formed is 12.
The maximum sum is 888.