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

If $33!$ is divisible by $2^n$,where $n \in N$,then the sum of all possible values of $n$ is:

$A$ box contains two white balls,three black balls,and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

Out of $18$ points in a plane,no three are in the same straight line except five points which are collinear. The number of $(i)$ straight lines,$(ii)$ triangles which can be formed by joining them is

There are $6$ multiple choice questions in an examination. How many sequences of answers are possible,if the first $2$ questions have $3$ choices each,the next $2$ have $4$ choices each,and the last $2$ have $5$ choices each?

In how many ways can $5$ boys and $5$ girls stand in a row so that no two girls may be together?

Vedclass Test Series

Exam Paper Generator

Online Exam Module