P and C doubt

Question

Number of ways in which the corner of the square can coloured with two colours.

It is permissible to use single colour on all four corners?

Comments

What is the given answer?

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6.......What u got

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It is permissible to use single colour on all four corners - given in the question

Answer 1

Case 1: Single colour use is permissible

In this case, you can visualize all corner as a 4 bit number arranged XXXX where each can be either 0 or 1. So you can have 2*2*2*2, total 16 possible ways.

Case 2: Both colour should be used

In this case just minus those two possible combination from total 16 combination where are colour of same type.

You will get 16 minus 2=14.

Let me know answer if you have.

Comments

Hey ! I got new insight. consider following two representation

XYXY

XXYY

I think in question they are not bothering about arranged of corners, so both of above two representation is same for them.  So I drawn a truth table of 16 entries and selected only those which are unique in representation irrespective of order. I got total 5 unique entries. Here I have consider that single colour is permissible.

Now If we can assume that no colour use is also permissible that it become 6.

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 0 as 1st color
 1 as 2nd color

0   0   0   0   |   0   
0   0   0   1   |   1  
0   0   1   0   |   2   
0   0   1   1   |   3   
0   1   0   0   |   4   
0   1   0   1   |   5  
0   1   1   0   |   6   
0   1   1   1   |   7   
1   0   0   0   |   8   
1   0   0   1   |   9   
1   0   1   0   |   10  
1   0   1   1   |   11  
1   1   0   0   |   12  
1   1   0   1   |   13  
1   1   1   0   |.  14
1   1   1   1   |.  15 

Considering two color and all vertices with same color also applicable

0000.       All with 0 color

1111.       All with 1 color

0111/1110/1101/1011     One 0 and  three 1 color

0011/0101/0110/1001/1010/1100       Two 0 and two 1 color

0001/0010/0100/1000        One 1 and three 1 color

I m getting only 5 ways only.

How we will get 6 ways:(

Plzz help..

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even I also told 5 only, Now 6 is possible when there is no colour..What i can uncesrtood