Question

$n$ distinct toys are placed in a row. In how many ways can a child select three toys so that no two of them are next to each other?

• A. ${}^n{C}_3$
• B. ${}^{n-2}{C}_3$
• C. ${}^{n-3}{C}_2$
• D. none of these

Answer 1

Answer: (B)

${}^{n-2}{C}_3$

Let $x_1=$ No. of toys between $1st$ toy $\xi$ $1st$ picked toy (including $1st$ toy ).

$x_2=$ No. of toys between $1st$ toy $\xi$ $2nd$ picked toy.

$x_3=$ No. of toys between $2nd$ toy $\xi$ $3rd$ picked toy.

$x_4=$ No. of toys remaining after the $3rd$ toy.

$\Rightarrow x_1+x_2+x_3+x_4+3=n$ $\xi$ $x_1,x_4\ge 0$

$x_2,x_3\ge 1$ (Since no $2$ toys should be next to each other)

Let $x_2=y_2+1$ $\to y_2\ge 0$

$x_3=y_3+1$ $\to y_3\ge 0$

$\Rightarrow x_1=y_1,x_4=y_4$

The equation becomes $y_1+y_2+y_3+y_4=n-3-2$

$\Rightarrow y_1+y_2+y_3+y_4=n-5$ $,$ $y_i\ge 0$

No of solutions $={}^{N+R-1}{C}_{R-1}$

$\Rightarrow N=n-5,R=4$

$\Rightarrow {}^{n-5+4-1}{C}_{4-1}={}^{n-2}{C}_3$

Hence, the answer is ${}^{n-2}{C}_3.$