Question

There are $12$ intermediate stations on a railway line between two stations. The number of ways that a train can be made to stop at $4$ of these intermediate stations, no two of these halting stations being consecutive is

• A. $125$
• B. $126$
• C. $127$
• D. $130$

Answer 1

Answer: (B)

$126$

Let $x_1,x_2,x_3,x_4,x_5$ be the no. of stations before first stop, between first and second stop so on after 4th stop then $x_1\ge 0,x_5\ge 0|x_2,x_3,x_4\ge 1$
and $x_1+x_2+x_3+x_4+x_5=12-4=8$
$x_1+x_2^1+x_3^1+x_4^1+x_5=8-3=5$
where $x_k^1\ge 0$
.$\cdot$. No. of non negative integral solutions is ${}^9{C}_4$
$=126$

Alternate Method:

This problem can be assumed as arranging $4$ identical oranges and $8$ identical apples in a linear way, such that no two oranges are together.

First arrange $8$ apples in $1$ way.

Now we have $9$ gaps between them, where we can place $4$ oranges in ${}^9{C}_4=126$ ways.