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Q3:
How many chords can be drawn through 21 points on a circle?
Solution :
Step 1: Theoretical Foundation
A chord of a circle is defined as a straight line segment whose endpoints both lie on the circumference of the circle. To form a unique chord, exactly two distinct points on the circle must be selected [Per the geometric definition of a chord]. Given a set of $n$ distinct points on a circle, the number of ways to choose $2$ points from the set of $n$ points is given by the combination formula:
$$^nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}$$
Where:
- $n = 21$ (Total number of points available)
- $r = 2$ (Number of points required to define a chord)
Step 2: Mathematical Derivation
Substituting the given variables into the combination formula:
$$\text{Number of chords} = ^{21}C_2 = \frac{21!}{2!(21-2)!}$$
Expanding the factorials for simplification:
$$\text{Number of chords} = \frac{21 \times 20 \times 19!}{2 \times 1 \times 19!}$$
Canceling the common $19!$ terms in the numerator and denominator [By the multiplicative property of factorials]:
$$\text{Number of chords} = \frac{21 \times 20}{2}$$
$$\text{Number of chords} = 21 \times 10 = 210$$
Step 3: Geometric Representation
Step 4: Analytical Conclusion
Since the points are on a circle, no three points are collinear. Therefore, every unique pair of points selected from the set of 21 will result in a unique chord. No geometric constraints (such as collinearity) reduce the total count calculated via the combination formula.
Final Solution: 210 chords
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.4
- Q1: If ${}^nC_8 = {}^nC_2$, find ${}^nC_2$.
- Q2(i): Determine $n$ if (i) ${}^{2n}C_3 : {}^nC_3 = 12 : 1$
- Q2(ii): Determine $n$ if (ii) ${}^{2n}C_3 : {}^nC_3 = 11 : 1$
- Q4: In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
- Q5: Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.
- Q6: Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
- Q7: In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?
- Q8: A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.
- Q9: In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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