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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?

Solution :

Step 1: Decomposition of the Given Word

The word EQUATION consists of $8$ distinct letters:

  • Vowels ($V$): {E, U, A, I, O} — Total $n_v = 5$
  • Consonants ($C$): {Q, T, N} — Total $n_c = 3$
[Per the Fundamental Principle of Counting, since the letters are distinct, we treat them as individual entities for arrangement.]

Step 2: Grouping Strategy

The problem requires vowels to occur together and consonants to occur together. We treat the set of vowels as one single unit ($V_{unit}$) and the set of consonants as one single unit ($C_{unit}$).

Total items to arrange = $\{V_{unit}, C_{unit}\}$.

Number of ways to arrange these 2 units = $2!$ [By the Permutation Theorem $n!$ for $n$ distinct items].

Step 3: Internal Permutations

A. Arranging Vowels internally: The 5 distinct vowels {E, U, A, I, O} can be arranged among themselves in $5!$ ways.

Calculation: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ ways.

B. Arranging Consonants internally: The 3 distinct consonants {Q, T, N} can be arranged among themselves in $3!$ ways.

Calculation: $3! = 3 \times 2 \times 1 = 6$ ways.

Step 4: Total Calculation

Applying the Multiplication Principle [If an event can occur in $m$ ways and another independent event in $n$ ways, the total number of occurrences is $m \times n$]:

Total Words = (Ways to arrange units) $\times$ (Internal arrangement of Vowels) $\times$ (Internal arrangement of Consonants)

Total Words = $2! \times 5! \times 3!$

Total Words = $2 \times 120 \times 6$

Total Words = $1440$

Vowels Group Consonants Group Arrangement: 2!

Final Solution: The total number of words that can be formed is $1440$.


More Questions from Class 11 Mathematics Permutations and Combinations Miscellaneous Exercise on Chapter 6


CBSE Solutions for Class 11 Mathematics Permutations and Combinations


Chapters in CBSE - Class 11 Mathematics


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