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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 ?
Solution :
Step 1: Identifying the Combinatorial Requirements
To form a word with $2$ distinct vowels and $2$ distinct consonants, we must execute the following logical sequence:
- Selection of vowels from the available pool.
- Selection of consonants from the available pool.
- Arrangement of the selected characters into a sequence (word).
Step 2: Selection of Characters
The total number of vowels is $n_v = 5$, and the number of consonants is $n_c = 21$. We are required to choose $r_v = 2$ vowels and $r_c = 2$ consonants.
The number of ways to select $2$ vowels from $5$ is given by the combination formula $^nC_r = \frac{n!}{r!(n-r)!}$:
$^5C_2 = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$The number of ways to select $2$ consonants from $21$ is:
$^{21}C_2 = \frac{21!}{2!(21-2)!} = \frac{21 \times 20}{2 \times 1} = 210$Total ways to select the $4$ letters ($2$ vowels and $2$ consonants) is: $10 \times 210 = 2100$ [By the Multiplication Principle of Counting]
Step 3: Arrangement of Selected Characters
Once $4$ distinct letters are selected, they can be arranged among themselves to form a word of length $4$. The number of ways to arrange $n$ distinct objects is given by $n!$:
$4! = 4 \times 3 \times 2 \times 1 = 24$Step 4: Final Calculation
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 them:
$\text{Total Words} = (\text{Ways to Select}) \times (\text{Ways to Arrange})$ $\text{Total Words} = 2100 \times 24$ $\text{Total Words} = 50400$Final Solution: 50,400 words
More Questions from Class 11 Mathematics Permutations and Combinations Miscellaneous Exercise on Chapter 6
- 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 ?
- 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 ?
- 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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