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Q6:
Find $n$ if ${}^{n–1}P_3 : {}^nP_4 = 1 : 9$.
Solution :
Step 1: Expanding the Permutation Formula
The formula for permutations of $n$ objects taken $r$ at a time is defined as:
${}^nP_r = \frac{n!}{(n-r)!}$ [Per the Fundamental Definition of Permutations]
We are given the ratio ${}^{n-1}P_3 : {}^nP_4 = 1 : 9$. Expressing this in fractional form:
$\frac{{}^{n-1}P_3}{{}^nP_4} = \frac{1}{9}$
Step 2: Applying the Permutation Formula to Given Terms
Substitute the specific values into the formula:
${}^{n-1}P_3 = \frac{(n-1)!}{(n-1-3)!} = \frac{(n-1)!}{(n-4)!}$
${}^nP_4 = \frac{n!}{(n-4)!}$
Step 3: Setting up the Algebraic Equation
Divide the two expressions:
$\frac{\frac{(n-1)!}{(n-4)!}}{\frac{n!}{(n-4)!}} = \frac{1}{9}$
By the algebraic property of complex fractions, since the denominators $(n-4)!$ are equivalent, they cancel out:
$\frac{(n-1)!}{n!} = \frac{1}{9}$
Step 4: Simplifying the Factorials
Recalling that $n! = n \times (n-1)!$ [By the Definition of Factorials], we substitute this into the denominator:
$\frac{(n-1)!}{n \times (n-1)!} = \frac{1}{9}$
Canceling the common term $(n-1)!$ from the numerator and denominator:
$\frac{1}{n} = \frac{1}{9}$
Step 5: Solving for $n$
By equating the reciprocal terms:
$n = 9$
Verification of Constraints
For the permutation ${}^nP_r$ to be defined, $n \geq r$.
- For ${}^{n-1}P_3$: $n-1 \geq 3 \implies n \geq 4$
- For ${}^nP_4$: $n \geq 4$
Since $n=9$ satisfies $n \geq 4$, the solution is valid within the domain of the permutation function.
Final Solution: $n = 9$
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?
- Q7(i): Find $r$ if (i) ${}^5P_r = 2 \cdot {}^6P_{r-1}$
- 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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