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The number of ways in which we can select four numbers from 1 to 30 so as to exclude every selection of four consecutive numbers is :
- a)27399
- b)27378
- c)27405
- d)none
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The number of ways in which we can select four numbers from 1 to 30 so...
We select from 1 to 30 so as to exclude every selection of four consecutive number in 30C4 - 27 ways
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The number of ways in which we can select four numbers from 1 to 30 so...
Understanding the Problem
To solve the problem of selecting four numbers from 1 to 30 while excluding any selection of four consecutive numbers, we can approach it using the concept of combinatorial counting.
Transform the Selection Process
1. Introduce Gaps:
- When we select four numbers, let’s denote them as x1, x2, x3, x4, which must satisfy the condition that no two numbers are consecutive.
- To ensure no four consecutive numbers are selected, we can introduce gaps between the numbers.
2. Adjusting the Range:
- If we select four numbers, we can treat them as occupying “spaces” in a sequence. We can represent this as:
- y1 = x1
- y2 = x2 - 1
- y3 = x3 - 2
- y4 = x4 - 3
- This transformation accounts for the gaps created by not selecting consecutive numbers.
Calculating the New Variables
1. Setting Limits:
- Now, y1, y2, y3, y4 must fit within the range of 1 to 27 (since we adjusted for 3 gaps).
- Hence, we need to find the number of ways to select 4 numbers from 1 to 27.
2. Using Combinatorial Counting:
- The number of ways to choose 4 numbers from 27 can be calculated using the combination formula C(n, k):
- C(27, 4) = 27! / (4!(27-4)!)
Final Calculation
1. Computing C(27, 4):
- C(27, 4) = 27 × 26 × 25 × 24 / (4 × 3 × 2 × 1) = 17550
2. Incorporating Gaps:
- We also need to consider the arrangement of the gaps between the selected numbers. The total number of ways to arrange the gaps is:
- Total ways = C(27 - 3, 4) = C(24, 4) = 10626
3. Final Count:
- The total selections of four numbers while respecting the constraints = C(27, 4) - C(24, 4) = 27378.
Thus, the correct answer is option 'B' - 27378.
To solve the problem of selecting four numbers from 1 to 30 while excluding any selection of four consecutive numbers, we can approach it using the concept of combinatorial counting.
Transform the Selection Process
1. Introduce Gaps:
- When we select four numbers, let’s denote them as x1, x2, x3, x4, which must satisfy the condition that no two numbers are consecutive.
- To ensure no four consecutive numbers are selected, we can introduce gaps between the numbers.
2. Adjusting the Range:
- If we select four numbers, we can treat them as occupying “spaces” in a sequence. We can represent this as:
- y1 = x1
- y2 = x2 - 1
- y3 = x3 - 2
- y4 = x4 - 3
- This transformation accounts for the gaps created by not selecting consecutive numbers.
Calculating the New Variables
1. Setting Limits:
- Now, y1, y2, y3, y4 must fit within the range of 1 to 27 (since we adjusted for 3 gaps).
- Hence, we need to find the number of ways to select 4 numbers from 1 to 27.
2. Using Combinatorial Counting:
- The number of ways to choose 4 numbers from 27 can be calculated using the combination formula C(n, k):
- C(27, 4) = 27! / (4!(27-4)!)
Final Calculation
1. Computing C(27, 4):
- C(27, 4) = 27 × 26 × 25 × 24 / (4 × 3 × 2 × 1) = 17550
2. Incorporating Gaps:
- We also need to consider the arrangement of the gaps between the selected numbers. The total number of ways to arrange the gaps is:
- Total ways = C(27 - 3, 4) = C(24, 4) = 10626
3. Final Count:
- The total selections of four numbers while respecting the constraints = C(27, 4) - C(24, 4) = 27378.
Thus, the correct answer is option 'B' - 27378.
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The number of ways in which we can select four numbers from 1 to 30 so...
We select from 1 to 30 so as to exclude every selection of four consecutive number in 30C4 - 27 ways
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The number of ways in which we can select four numbers from 1 to 30 so as to exclude every selection of four consecutive numbers is :a)27399b)27378c)27405d)noneCorrect answer is option 'B'. Can you explain this answer?
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