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There are 12 intermediate stations between two places A and B. In how many ways can a train be made to stop at 4 of these 12 intermediate stations that no two stations are consecutive?
- a)15C3
- b)11C3
- c)9C4
- d)9C3
Correct answer is option 'C'. Can you explain this answer?
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There are 12 intermediate stations between two places A and B. In how ...
_S1_S2_S3_S4_ ----> there should minimum 3 stations b/w the 4 stations. therefore 12 - 3 = 9 points where train may be stopped. --> 9C4 is the answer.
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There are 12 intermediate stations between two places A and B. In how ...
Problem: To find the number of ways to select 4 non-consecutive stations from 12 intermediate stations between A and B.
Solution:
Method 1: Using Complementary Counting
We can find the required number of ways by subtracting the number of ways to select 4 consecutive stations from the total number of ways to select any 4 stations.
Total number of ways to select 4 stations from 12 intermediate stations = 12C4
Number of ways to select 4 consecutive stations = 9 (since there are 9 possible sets of 4 consecutive stations)
Therefore, number of ways to select 4 non-consecutive stations = 12C4 - 9 = 495 - 9 = 486
Method 2: Using Counting Principles
We can also use counting principles to directly count the number of ways to select 4 non-consecutive stations.
Step 1: Select any 4 stations from the 12 intermediate stations (without any restrictions). This can be done in 12C4 ways.
Step 2: Among the selected 4 stations, there are 3 pairs of consecutive stations. We need to exclude all such selections.
Step 3: To exclude the selections with consecutive stations, we can select any one of the 3 pairs of consecutive stations and remove one station from each pair. This will give us a selection of 4 non-consecutive stations.
Step 4: The number of ways to select a pair of consecutive stations is 9 (there are 9 pairs of consecutive stations). For each such pair, there are 2 ways to remove one station (either the first or the second station in the pair).
Therefore, the total number of ways to select 4 non-consecutive stations = 12C4 - 9*2 = 495 - 18 = 477
Answer: Option (c) 9C4 is the correct answer.
Solution:
Method 1: Using Complementary Counting
We can find the required number of ways by subtracting the number of ways to select 4 consecutive stations from the total number of ways to select any 4 stations.
Total number of ways to select 4 stations from 12 intermediate stations = 12C4
Number of ways to select 4 consecutive stations = 9 (since there are 9 possible sets of 4 consecutive stations)
Therefore, number of ways to select 4 non-consecutive stations = 12C4 - 9 = 495 - 9 = 486
Method 2: Using Counting Principles
We can also use counting principles to directly count the number of ways to select 4 non-consecutive stations.
Step 1: Select any 4 stations from the 12 intermediate stations (without any restrictions). This can be done in 12C4 ways.
Step 2: Among the selected 4 stations, there are 3 pairs of consecutive stations. We need to exclude all such selections.
Step 3: To exclude the selections with consecutive stations, we can select any one of the 3 pairs of consecutive stations and remove one station from each pair. This will give us a selection of 4 non-consecutive stations.
Step 4: The number of ways to select a pair of consecutive stations is 9 (there are 9 pairs of consecutive stations). For each such pair, there are 2 ways to remove one station (either the first or the second station in the pair).
Therefore, the total number of ways to select 4 non-consecutive stations = 12C4 - 9*2 = 495 - 18 = 477
Answer: Option (c) 9C4 is the correct answer.
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