The number of four-letter words that can be f | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The word $BARRACK$ contains $7$ letters: $A, A, R, R, B, C, K$. The distinct letters are ${A, R, B, C, K}$.Case $1$: All $4$ letters are distinct.

Vedclass · Accepted Answer

$270$

Vedclass · Answer

(D) The word $BARRACK$ contains $7$ letters: $A, A, R, R, B, C, K$. The distinct letters are ${A, R, B, C, K}$.Case $1$: All $4$ letters are distinct.We choose $4$ letters from ${A, R, B, C, K}$ in $^5C_4 = 5$ ways. These $4$ letters can be arranged in $4! = 24$ ways.Total words $= 5 	imes 24 = 120$.Case $2$: Two letters are the same (a pair) and two are distinct.Subcase $2a$: The pair is $R, R$. We choose $2$ other distinct letters from ${A, B, C, K}$ in $^4C_2 = 6$ ways. The number of arrangements is $\frac{4!}{2!} = 12$ ways.Total words $= 6 	imes 12 = 72$.Subcase $2b$: The pair is $A, A$. We choose $2$ other distinct letters from ${R, B, C, K}$ in $^4C_2 = 6$ ways. The number of arrangements is $\frac{4!}{2!} = 12$ ways.Total words $= 6 	imes 12 = 72$.Case $3$: Two pairs of letters ($A, A$ and $R, R$).We choose $2$ pairs from ${A, A, R, R}$ in $^2C_2 = 1$ way. The number of arrangements is $\frac{4!}{2!2!} = 6$ ways.Total words $= 1 	imes 6 = 6$.Total number of words $= 120 + 72 + 72 + 6 = 270$.

The number of four-letter words that can be formed using the letters of the word $BARRACK$ is

Similar Questions

In how many different ways can the letters of the word '$BANKING$' be arranged so that the vowels always come together?

The number of arrangements of the letters of the word $PALANHAR$ in which no two vowels are together and exactly two vowels are at odd places,is

The Chief Ministers of $11$ States of India meet to discuss the language problem. In how many ways can they seat themselves at a round table so that the Punjab and Madras Chief Ministers sit together?

Consider all possible permutations of the letters of the word $EARTHQUAKE$. The number of permutations containing the word $RAHU$ is:

If $P(56, r+6) : P(54, r+3) = 30800 : 1$,find $r$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module