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Q4:
In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
Solution :
Step 1: Theoretical Foundation & Formulation
To determine the number of ways to select a specific committee or team where the order of selection is irrelevant, we employ the Combination formula. The number of ways to select $r$ items from a set of $n$ distinct items is defined as:
$^{n}C_{r} = \frac{n!}{r!(n-r)!}$
[Per the Fundamental Principle of Combinatorics for selection without arrangement]
Step 2: Decomposition of Selection Tasks
The problem requires two independent selection processes to be performed simultaneously to form the team:
- Task A: Selecting 3 boys from a group of 5 boys.
- Task B: Selecting 3 girls from a group of 4 girls.
Step 3: Algebraic Calculation of Selections
Applying the formula to Task A ($n=5, r=3$):
$^{5}C_{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = \frac{20}{2} = 10$
Applying the formula to Task B ($n=4, r=3$):
$^{4}C_{3} = \frac{4!}{3!(4-3)!} = \frac{4 \times 3!}{3! \times 1!} = 4$
Step 4: Applying the Fundamental Counting Principle
According to the Fundamental Principle of Multiplication, if one operation can be performed in $m$ ways and a second independent operation in $n$ ways, the total number of ways to perform both operations is $m \times n$.
Total ways = ($^{5}C_{3}$) $\times$ ($^{4}C_{3}$)
Total ways = $10 \times 4 = 40$
Visual Representation of Selection Logic
Final Solution: The total number of ways to select the team is 40 ways.
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?
- 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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