How many chords can be drawn through 21 points on a circle?
A chord of a circle is a straight line segment whose end points lie on the circle. Since only 2 points are required to draw one chord, so the number of chords drawn from the 21 points on the circle is the number of combinations that have to be counted.
The number of chords is the combination of 21 points taken 2 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r !
Substitute 21 for n and 2 for r in the above formula.
C 21 2 = 21 ! ( 21 − 2 ) ! 2 ! = 21 ! 19 ! 2 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 21 2 = 21 × 20 × 19 ! 19 ! × 2 × 1 = 21 × 20 2 × 1 = 210
Thus, the required number of chords is 210.
A chord of a circle is a straight line segment whose end points lie on the circle. Since only 2 points are required to draw one chord, so the number of chords drawn from the 21 points on the circle is the number of combinations that have to be counted.
The number of chords is the combination of 21 points taken 2 at a time.
The formula for the combination is defined as,
Substitute 21 for n and 2 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
Thus, the required number of chords is 210.
