If x and y represent the number of arrangemen | vedclass

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

Question

(C) The word $ATRAPATRAM$ contains $10$ letters: $A(4), T(2), R(2), P(1), M(1)$.$(i)$ If all $A$'s are together,we treat the block $(AAAA)$ as one uni

Vedclass · Accepted Answer

$\frac{6!}{2!2!} 	imes 42$

Vedclass · Answer

(C) The word $ATRAPATRAM$ contains $10$ letters: $A(4), T(2), R(2), P(1), M(1)$.$(i)$ If all $A$'s are together,we treat the block $(AAAA)$ as one unit. We then have $6$ other letters plus this $1$ unit,totaling $7$ items. The number of arrangements is $x = \frac{7!}{2!2!}$.(ii) If no two $A$'s are together,we first arrange the remaining $6$ letters $(T, T, R, R, P, M)$,which can be done in $\frac{6!}{2!2!}$ ways. There are $7$ gaps created by these $6$ letters. We choose $4$ gaps for the $4$ $A$'s in $^7C_4$ ways. Thus,$y = ^7C_4 	imes \frac{6!}{2!2!}$.Therefore,$x+y = \frac{7!}{2!2!} + ^7C_4 	imes \frac{6!}{2!2!}$.$x+y = \frac{7 	imes 6!}{2!2!} + 35 	imes \frac{6!}{2!2!} = \frac{6!}{2!2!} (7 + 35) = \frac{6!}{2!2!} 	imes 42$.

If $x$ and $y$ represent the number of arrangements of the letters of the word $ATRAPATRAM$ such that $(i)$ all $A$'s are together and $(ii)$ no two $A$'s are together respectively,then $x+y$ is equal to:

Similar Questions

The number of three-digit numbers in which $9$ appears in exactly one place is

Let $5$-digit numbers be constructed using the digits $0, 2, 3, 4, 7, 9$ with repetition allowed,and are arranged in ascending order with serial numbers. Then the serial number of the number $42923$ is $...............$.

The number of $9$-digit even natural numbers formed using only the digits $0$ and $1$,such that no two consecutive digits are $0$,is:

If the number of all possible permutations of the letters of the word $MATHEMATICS$ in which the repeated letters are not together is $90(X)$,then $X=$

The total number of two-digit numbers $n$ such that $3^{n} + 7^{n}$ is a multiple of $10$ is ..... .

Vedclass Test Series

Exam Paper Generator

Online Exam Module