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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.
Solution :
Step 1: Analyzing the Selection Parameters
To determine the number of ways to select balls of specific colors, we employ the principle of combinations, as the order of selection does not matter [Per the definition of combinations: $^nC_r = \frac{n!}{r!(n-r)!}$].
Given parameters:
- Total number of black balls ($n_b$) = $5$
- Number of black balls to be selected ($r_b$) = $2$
- Total number of red balls ($n_r$) = $6$
- Number of red balls to be selected ($r_r$) = $3$
Step 2: Calculating Combinations for Each Color
First, calculate the number of ways to select 2 black balls from 5 available [By the Combination Formula]:
$$^5C_2 = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10$$
Second, calculate the number of ways to select 3 red balls from 6 available:
$$^6C_3 = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4 \times 3!}{ (3 \times 2 \times 1) \times 3!} = \frac{120}{6} = 20$$
Step 3: Applying the Fundamental Principle of Counting
Since the selection of black balls and the selection of red balls are independent events that must both occur to satisfy the condition, we use the Fundamental Principle of Multiplication [Per the Fundamental Counting Principle (Multiplication Rule)]:
$$\text{Total Ways} = (^5C_2) \times (^6C_3)$$
$$\text{Total Ways} = 10 \times 20 = 200$$
Final Solution: The total number of ways to select the balls is 200.
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) ${}^{2n}C_3 : {}^nC_3 = 11 : 1$
- 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?
- 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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