Let S=1,2,3,4,5,6,7,8,9.

Question

Let $S={1,2,3,4,5,6,7,8,9}$. Let $x$ be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9 -digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

Accepted Answer

Given, $S={1,2,3,4,5,6,7,8,9}$

Calculation for x

For $x$, we have to form a 9-digit number in which exactly one digit is repeated, and it is repeated exactly twice. So, one digit appears twice, and the remaining 7 digits appear once each.

First, select the digit that will be repeated: from $S$, this can be done in ${}^{9}{C}_{1}$ ways.

Now, from the remaining 8 digits, choose 7 digits that will appear once: this can be done in ${}^{8}{C}_{7}$ ways.

Now we have 9 digits in total, but one digit is repeated twice. So, the number of different arrangements is $=\frac{9!}{2!}$ ways.

$x={}^{9}{C}_{1}	imes {}^{8}{C}_{7}	imes \frac{9!}{2!}$

Calculation for y

For $y$, we have to form a 9-digit number in which exactly two digits are repeated, and each of these is repeated exactly twice. So, two digits appear twice each (total 4 places), and the remaining 5 digits appear once each.

First, select the 2 digits that will be repeated from $S$: this can be done in ${}^{9}{C}_{2}$ ways.

Now, from the remaining 7 digits, choose 5 digits that will appear once: this can be done in ${}^{7}{C}_{5}$ ways.

Now we have 9 digits in total, with two digits repeated twice each. So, the number of different arrangements is $=\frac{9!}{2!2!}$ ways.

$y={}^{9}{C}_{2}	imes {}^{7}{C}_{5}	imes \frac{9!}{2!2!}$

To find the relation between $x$ and $y$, divide $x$ by $y$:

To find the relation, divide x by y :

$\frac{x}{y}=\frac{{}^{9}{C}_{1}	imes {}^{8}{C}_{7}	imes \frac{1}{2}}{{}^{9}{C}_{2}	imes {}^{7}{C}_{5}	imes \frac{1}{4}}=\frac{9	imes 8	imes \frac{1}{2}}{36	imes 21	imes \frac{1}{4}}$

So,

$\frac{x}{y}=\frac{36}{189}=\frac{4}{21}$

$\Rightarrow 21x=4y$

Solutions · 1