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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?
Solution :
Step 1: Identifying the Set of Letters
The word given is MONDAY. We must first identify the constituent letters to determine the total count and the subsets of vowels and consonants.
- Total letters: $M, O, N, D, A, Y$ (Total = $6$ unique letters).
- Vowels: $O, A$ (Total = $2$).
- Consonants: $M, N, D, Y$ (Total = $4$).
Step 2: Defining the Positional Constraints
We are forming a $6$-letter word using all letters of MONDAY without repetition, subject to the condition that the first letter must be a vowel.
Let the positions be represented as $P_1, P_2, P_3, P_4, P_5, P_6$.
Constraint: $P_1 \in \{O, A\}$.
Step 3: Calculating Permutations for the Restricted Position
Since the first position ($P_1$) must be filled by a vowel, and there are $2$ available vowels ($O$ or $A$), the number of ways to fill the first position is:
$N(P_1) = 2$ [By Fundamental Counting Principle: Multiplication Rule].
Step 4: Calculating Permutations for the Remaining Positions
After selecting $1$ vowel for the first position, we are left with $5$ remaining letters to be arranged in the remaining $5$ positions ($P_2, P_3, P_4, P_5, P_6$).
The number of ways to arrange $5$ distinct items in $5$ spots is given by $5!$ [Per the permutation formula $^nP_r = \frac{n!}{(n-r)!}$ where $n=5, r=5$].
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
Step 5: Final Computation
Total number of words = (Ways to choose $P_1$) $\times$ (Ways to arrange remaining letters).
Total = $2 \times 120 = 240$.
Final Solution: 240 words
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(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,
- 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?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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