## mathmath333 one year ago Counting Problem

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{There are \(10 points in a plane of which $$4$$ are collinear. }\hspace{.33em}\\~\\ & \normalsize \text{No other $$3$$ points are collinear.}\hspace{.33em}\\~\\ & \normalsize \text{Find the number of lines that can be drawn .}\hspace{.33em}\\~\\ \end{align}}\)

2. ganeshie8

is the answer $$\dbinom{6}{2} + 6*4$$ ?

3. mathmath333

10C2-4C2+1

4. ganeshie8

right, both are same just a +1 is missing in my earlier reply

5. mathmath333

how u got that

6. ganeshie8

7. ganeshie8

then you have 6 points, you know that any two points determine a line so how many lines can you make using 6 points ?

8. mathmath333

6c2

9. ganeshie8

next, from each of these 6 points, you can draw a line to each of the 4 collinear points, yes ?

10. mathmath333

yes

11. ganeshie8

|dw:1440183836929:dw|

12. ganeshie8

how many lines can you draw from those 6 points to those 4 collinear points ?

13. mathmath333

6C4

14. ganeshie8

|dw:1440183971999:dw|

15. ganeshie8

nope, notice that from each of those 6 points, you can draw exactly 4 lines to those collinear points

16. mathmath333

4*6

17. ganeshie8

Yes, so far the count is $$\dbinom{6}{2}+4*6$$ can you guess the one line that is missing ?

18. mathmath333

|dw:1440184361598:dw|

19. ganeshie8

Thats it!

20. ganeshie8

so the final answer is $$\dbinom{6}{2} + 6*4+1$$

21. mathmath333

ok thnsk

22. ganeshie8