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Q1:
If ${}^nC_8 = {}^nC_2$, find ${}^nC_2$.
Solution :
Step 1: Application of the Complementary Combination Theorem
The problem presents the equation ${}^nC_8 = {}^nC_2$. We utilize the fundamental theorem of combinations which states that if ${}^nC_a = {}^nC_b$, then either $a = b$ or $n = a + b$.
[Per the property: ${}^nC_r = {}^nC_{n-r}$]
Since $8 \neq 2$, we must satisfy the condition:
$n = 8 + 2$
$n = 10$
Step 2: Verification of Variable $n$
With $n = 10$, we check the equality:
${}^{10}C_8 = \frac{10!}{8!(10-8)!} = \frac{10 \times 9}{2 \times 1} = 45$
${}^{10}C_2 = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45$
[Verified: ${}^{10}C_8 = {}^{10}C_2$ holds true]
Step 3: Calculating the Required Value ${}^nC_2$
Substituting $n = 10$ into the expression ${}^nC_2$:
${}^{10}C_2 = \frac{n!}{r!(n-r)!} = \frac{10!}{2!(10-2)!}$
${}^{10}C_2 = \frac{10 \times 9 \times 8!}{2 \times 1 \times 8!}$
${}^{10}C_2 = \frac{90}{2} = 45$
Visual Representation of Combinatorial Distribution
Final Solution: ${}^{n}C_2 = 45$
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.4
- Q2(i): Determine $n$ if (i) ${}^{2n}C_3 : {}^nC_3 = 12 : 1$
- Q2(ii): Determine $n$ if (ii) ${}^{2n}C_3 : {}^nC_3 = 11 : 1$
- Q3: How many chords can be drawn through 21 points on a circle?
- 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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