Can you solve this?

Number Theory Level 3

In how many ways can three different integers be selected from the numbers 1 to 12,so that their sum can be exactly divided by 3?

76 56 84 64

Anik Mandal by Anik Mandal , 21, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Mathh Mathh
Jul 31, 2014

Those numbers can be divided into groups ( m o d 3 ) \pmod 3 :

{ 1 − 1 0 1 − 1 0 1 − 1 0 1 − 1 0 \begin{cases}1 &&-1 && 0\\1 &&-1 && 0\\1 &&-1 && 0\\1 &&-1 && 0\end{cases}

We can either choose 3 ones, 3 minus ones, 3 zeros or 1 of each group. ( 4 3 ) + ( 4 3 ) + ( 4 3 ) + ( 4 1 ) ( 4 1 ) ( 4 1 ) = 76 {4\choose 3}+{4\choose 3}+{4\choose 3}+{4\choose 1}{4\choose 1}{4\choose 1}=\boxed{76}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Good Solution!!

Anik Mandal - 6 years, 10 months ago

Follow me i will follow you back.

Sourabh Nolkha - 6 years, 10 months ago

Log in to reply

Who are you referring to? @Sourabh Nolkha

Anik Mandal - 6 years, 10 months ago
Adarsh Kumar
Aug 1, 2014

The numbers from 1 to 12 leave remainders 0,1,2 when divided by 3.If the sum of 3 numbers is divisible by 3 the numbers can have residues 0,0,0 1,1,1 2,2,2 and 0,1,2.Now do it with the help of combinatorics.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...