There are 15 stations on a train route and the train has to be stopped at exactly 5 stations among these 15 stations. If it stops at at least two consecutive stations, then the number of ways in which the train can be stopped is
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When a counting problem involves phrases like "at least one" or "at least two," it's often easier to count the total number of cases and subtract the number of cases where the condition is not met (the complement). The formula $^{n-k+1}C_k$ for selecting $k$ non-consecutive objects from $n$ is a very useful shortcut.
Step 1: Total Ways
Number of ways to choose 5 stations out of 15 is \( ^{15}C_5 \).
Step 2: Complementary Counting
Condition: Stops at "at least two consecutive stations".
Complement: Stops such that "no two stations are consecutive".
Step 3: Calculate Complement
The number of ways to choose \( k \) items from \( n \) such that no two are consecutive is given by \( ^{n-k+1}C_k \).
Here, \( n=15, k=5 \).
Ways = \( ^{15-5+1}C_5 = ^{11}C_5 \).
Step 4: Final Result
Required Ways = Total Ways - Complement Ways
\( = ^{15}C_5 - ^{11}C_5 \).