Question

Consider all the different words that can be formed using the letters of the word HAVANA, taken 4 at a time.

Answer 1

$HAVANA$

No of distinct letters =$4(H,A,V,N)4$

No of letters repeating thrice = $1\left(A\right)$

(A) No of words with all letters different ${=}^4{C}_4\times 4!=24$

(B) No of words with $2$ alike and $2$ different letters

Here $2$ alike letters we can not have $A$

again So, required answer

${=}^3{C}_2{\times }^1{C}_1\times \frac{4!}{2!}=3\times 12=36\downarrow \searrow ChoosingChoosingany2letters2lettersfromH,V,NfromA$

(c) No of words in which A's never appear together.

cases:-(1) If all letters are different then A's will never appear together

(2)If we have 2 A's then we need to remove the cases in which A's appear together

(3)If we have 3 A's then we will always have A's together

$\therefore Reqanswer=24{+}^3{C}_2{\times }^1{C}_1[\frac{4!}{2!}-3!]+0\downarrow \downarrow \downarrow \left(1\right)\left(2\right)\left(3\right)=24+3\left(6\right)=42$

(d)

No of words starting with $HAA$

$HAA\_\hookrightarrow 3possibilities$

$\therefore Total=33+3=36$

Now, the next word is$HANA$.

Hence, rank of $HANA$ is $36+1={37}^{th}$