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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?
Solution :
Given Variables & Initial Setup
- Total number of courses available ($n_{total}$) = $9$
- Number of courses to be chosen ($r_{total}$) = $5$
- Number of specific compulsory courses ($n_{comp}$) = $2$
Since the 2 specific courses are compulsory, they must be included in every student's selection. This effectively reduces the number of courses the student needs to "choose" and the number of "options" available for selection.
Step 1: Calculating Remaining Selections
Because $2$ courses are already fixed, the number of courses remaining to be chosen is:
$r_{rem} = r_{total} - n_{comp} = 5 - 2 = 3$
[By the fundamental principle of subtraction in selection processes]
Step 2: Calculating Remaining Options
The number of available courses from which the student can make the remaining selection is:
$n_{rem} = n_{total} - n_{comp} = 9 - 2 = 7$
Step 3: Combinatorial Calculation
The problem now reduces to selecting $3$ courses out of the remaining $7$ available courses. Since the order of selection does not matter for a programme of courses, we use the combination formula $^nC_r = \frac{n!}{r!(n-r)!}$ [Per the definition of combinations].
Substituting the values:
$^7C_3 = \frac{7!}{3!(7-3)!}$
$^7C_3 = \frac{7!}{3! \times 4!}$
$^7C_3 = \frac{7 \times 6 \times 5 \times 4!}{ (3 \times 2 \times 1) \times 4!}$
$^7C_3 = \frac{7 \times 6 \times 5}{6}$
$^7C_3 = 7 \times 5 = 35$
Visual Representation of Selection Logic
Comparison Table of Selection Logic
| Category | Value |
|---|---|
| Total Pool | 9 |
| Fixed Selection | 2 |
| Remaining Pool | 7 |
| Remaining Slots | 3 |
Final Solution: 35 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?
- 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.
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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