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Q7(i):
Find $r$ if (i) ${}^5P_r = 2 \cdot {}^6P_{r-1}$
Solution :
Step 1: Theoretical Foundation
The permutation formula is defined as ${}^nP_r = \frac{n!}{(n-r)!}$, where $0 \leq r \leq n$ [Per the fundamental principle of permutations].
Step 2: Substitution into the Given Equation
Given equation: ${}^5P_r = 2 \cdot {}^6P_{r-1}$
Expanding both sides using the permutation formula:
$\frac{5!}{(5-r)!} = 2 \cdot \frac{6!}{(6-(r-1))!}$
$\frac{5!}{(5-r)!} = 2 \cdot \frac{6!}{(7-r)!}$
Step 3: Algebraic Simplification
Expand the factorial terms to identify common factors:
$\frac{5!}{(5-r)!} = 2 \cdot \frac{6 \cdot 5!}{(7-r)(6-r)(5-r)!}$
Cancel $5!$ from both sides and $(5-r)!$ from the denominators:
$1 = \frac{2 \cdot 6}{(7-r)(6-r)}$
$1 = \frac{12}{(7-r)(6-r)}$
Step 4: Solving the Quadratic Equation
Rearrange to form a standard quadratic equation:
$(7-r)(6-r) = 12$
$42 - 7r - 6r + r^2 = 12$
$r^2 - 13r + 42 - 12 = 0$
$r^2 - 13r + 30 = 0$
Factorize the quadratic expression:
$(r - 10)(r - 3) = 0$
Step 5: Verification of Constraints
The roots are $r = 10$ and $r = 3$.
- If $r = 10$: The term ${}^5P_{10}$ is undefined because $n < r$ ($5 < 10$). [Per the domain constraint $r \leq n$].
- If $r = 3$: Check feasibility: ${}^5P_3 = \frac{5!}{2!} = 60$. Right side: $2 \cdot {}^6P_{3-1} = 2 \cdot {}^6P_2 = 2 \cdot \frac{6!}{4!} = 2 \cdot (6 \cdot 5) = 60$.
The condition $60 = 60$ holds true.
Final Solution: r = 3
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.3
- Q1: How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
- Q2: How many 4-digit numbers are there with no digit repeated?
- Q3: How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?
- Q4: Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?
- Q5: From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person can not hold more than one position?
- Q6: Find $n$ if ${}^{n–1}P_3 : {}^nP_4 = 1 : 9$.
- Q7(ii): Find $r$ if (ii) ${}^5P_r = {}^6P_{r-1}$.
- Q8: How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?
- Q9(i): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (i) 4 letters are used at a time,
- Q9(ii): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (ii) all letters are used at a time,
- Q9(iii): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (iii) all letters are used but first letter is a vowel?
- Q10: In how many of the distinct permutations of the letters in MISSISSIPPI do the four $I$'s not come together?
- Q11(i): In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) words start with P and end with S,
- Q11(ii): In how many ways can the letters of the word PERMUTATIONS be arranged if the (ii) vowels are all together,
- Q11(iii): In how many ways can the letters of the word PERMUTATIONS be arranged if the (iii) there are always 4 letters between P and S?
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