Question

If 5 vowels and 6 consonants are given, then how many 6 letter words can be formed with 3 vowels and 3 consonants?

Answer 1

Hint: First we will find the number of ways to choose vowels and consonants separately by using the formula of combination, which is given by
$${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}$$
Where, $n=$ number of items/objects
And $r=$ number of items/objects being chosen at a time
Then, we find the number of ways to choose both vowels and consonants. Then, find the number of ways to arrange them to form 6 letter words. Then, multiply the obtained numbers to get the desired result.

Complete step by step answer:
We have given 5 vowels and 6 consonants.
Then, we have to find how many 6 letter words can be formed with 3 vowels and 3 consonants.
Now, we need to choose 3 vowels from the given 5 vowels. So, the number of ways to choose vowels will be
$${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}$$
$$\begin{array}{rl}& \Rightarrow {}^5{C}_3={\displaystyle \frac{5!}{3!(5-3)!}}\\ & \Rightarrow {}^5{C}_3={\displaystyle \frac{5\times 4\times 3!}{3!\left(2\right)!}}\\ & \Rightarrow {}^5{C}_3={\displaystyle \frac{5\times 4}{2\times 1}}\\ & \Rightarrow {}^5{C}_3={\displaystyle \frac{20}{2}}\\ & \Rightarrow {}^5{C}_3=10\end{array}$$
Now, we have to find the number of ways to choose 3 consonants from the given 6 consonants, we get
$${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}$$
$$\begin{array}{rl}& \Rightarrow {}^6{C}_3={\displaystyle \frac{6!}{3!(6-3)!}}\\ & \Rightarrow {}^6{C}_3={\displaystyle \frac{6\times 5\times 4\times 3!}{3!\left(3\right)!}}\\ & \Rightarrow {}^6{C}_3={\displaystyle \frac{6\times 5\times 4}{3\times 2\times 1}}\\ & \Rightarrow {}^6{C}_3={\displaystyle \frac{120}{6}}\\ & \Rightarrow {}^6{C}_3=20\end{array}$$
Now, we have to find the number of ways to select both vowels and consonants.
We have $${}^5{C}_3=10$$ and $${}^6{C}_3=20$$, so number of ways to choose both will be
$$\begin{array}{rl}& {}^5{C}_3\times {}^6{C}_3=20\times 10\\ & {}^5{C}_3\times {}^6{C}_3=200\end{array}$$
Now, we have to form 6 letter words by arranging this vowels and consonants. So, 6 letters are arranged in $6!$ ways.
So, the total number of words that can be formed will be
$\begin{array}{rl}& 6!\times 200\\ & =6\times 5\times 4\times 3\times 2\times 1\times 200\\ & =144000\end{array}$

So, total $144000$ words can be formed.

Note: There is a possibility that students may forget to arrange 6 letters to form a word and give the answer as $200$, which is an incorrect answer. Each arrangement of 6 letters gives different words so it is necessary to arrange them within themselves. Also, there is a difference between permutation and combination. Combination means only choosing while permutation means first choosing then arranging.