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→Hints for difficult problems
= Hints for difficult problems =
= Hints for difficult problems =
# How many 1) lines`
2) Triangles can be drawn through 8 points on a circle
Interpetation:
drawing lines and triangle using the 8 points which are on a circumference of a circle (irrespective of the position on circumference , may be equally spaced or not) . And students should identify whether the problem is of arrengement or selection of objects.
Methods:
1) visual method
2) arithmetic way
Concepts:
1. circle
2. co-linear and non co-linear points
3. difference bet'n line, ray and segment
4. points require to draw a triangle
5. difference bet'n arrengements and selection for eg. If A and B are two objects then AB and BA are different arregements where as AB and BA are both same when it comes to selection.
6. Expantion of Factorial
Solution:
1. for drawing lines we need atleast two points then selecting 2 points out of 8 given we have
<br> = arrangements
<br>n!
<br> =----------- ( students should know the formula)
<br>(n-r)! r!
<br> 8!
<br> = ----------- (substituting the values into the formula)
<br> (8-2)! 2!
<br>8!
<br>= ----------- (simplification)
<br>6! 2!
<br>
<br> 8*7*6*5*4*3*2*1
<br> =--------------------------
<br> 6*5*4*3*2*1*2*1
<br> =4*7
<br>
<br> = 28
<br> 2. for drawing a triangle we need 3 non co-linear points then selecting 3 points out of 8 given we have
<br> = arrengements
<br> n!
<br> =----------- ( students should know the formula)
<br> (n-r)! r!
<br> 8!
<br> = ----------- (substituting the values into the formula)
<br> (8-3)! 3!
<br> 8!
<br>= ----------- (simplification)
<br>5! 3!
<br>
<br> 8*7*6*5*4*3*2*1
<br> =--------------------------
<br> 5*4*3*2*1*3*2*1
<br> =8*7
<br> = 56
= Project Ideas =
= Project Ideas =