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Q1:
How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER ?
Solution :
Step 1: Analyzing the Source Word
The word "DAUGHTER" consists of the following 8 unique letters: {D, A, U, G, H, T, E, R}.
- Vowels in the set: {A, U, E} (Total count = 3)
- Consonants in the set: {D, G, H, T, R} (Total count = 5)
Step 2: Selection Process (Combinations)
To form a word with 2 vowels and 3 consonants, we must perform two independent selection tasks [Per the Fundamental Counting Principle of Selection]:
- Selecting 2 vowels from 3: $^3C_2 = \frac{3!}{2!(3-2)!} = 3$ ways.
- Selecting 3 consonants from 5: $^5C_3 = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10$ ways.
Total combinations of letters = $3 \times 10 = 30$ ways.
Step 3: Arrangement Process (Permutations)
Each set of 5 selected letters can be arranged among themselves to form different words. The number of permutations of $n$ distinct objects is given by $n!$ [Per the Permutation Theorem].
For each selection of 5 letters, the number of ways to arrange them is $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ ways.
Step 4: Total Permutations Calculation
Using the Multiplication Principle for counting, the total number of words that can be formed is the product of the number of ways to select the letters and the number of ways to arrange those selected letters:
Total Words = $(\text{Ways to Select}) \times (\text{Ways to Arrange})$
Total Words = $30 \times 120 = 3,600$.
Final Solution: 3,600 words
More Questions from Class 11 Mathematics Permutations and Combinations Miscellaneous Exercise on Chapter 6
- Q2: How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?
- Q3(i): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls ?
- Q3(ii): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (ii) atleast 3 girls ?
- Q3(iii): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (iii) atmost 3 girls ?
- Q4: If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E ?
- Q5: How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated ?
- Q6: The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet ?
- Q7: In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions ?
- Q8: Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.
- Q9: It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible ?
- Q10: From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen ?
- Q11: In how many ways can the letters of the word ASSASSINATION be arranged so that all the $S$'s are together ?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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