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The number of selections of n different things taken r at a time which exclude a particular thing is
- a)nC(n-1,r-1)
- b)C(n-1,r)
- c)rC(n-1,r)
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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The number of selections of n different things taken r at a time which...
Solution:
The number of selections of n different things taken r at a time can be calculated using the combination formula, denoted as "nCr" or "C(n,r)". The formula is given by:
C(n,r) = n! / (r! * (n-r)!)
Here, n! represents the factorial of n, which is the product of all positive integers less than or equal to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120.
Now, let's consider the scenario where we want to select r things from n different things, excluding a particular thing. In this case, we have (n-1) things to choose from as we are excluding one thing.
Explanation:
To solve this problem, we need to find the number of selections of (n-1) different things taken r at a time.
Using the combination formula, we can write this as:
C(n-1, r) = (n-1)! / (r! * ((n-1)-r)!)
Simplifying this expression, we get:
C(n-1, r) = (n-1)! / (r! * (n-r-1)!)
Now, let's compare this expression with the given options:
a) nC(n-1, r-1) = n! / ((r-1)! * (n-r)!)
b) C(n-1, r) = (n-1)! / (r! * (n-r-1)!)
c) rC(n-1, r) = r! * ((n-1)! / ((r-1)! * (n-r)!)
d) none of these
Comparing option b) with the derived expression, we can see that they are the same. Therefore, the correct answer is option b) - C(n-1, r).
Summary:
The number of selections of n different things taken r at a time, excluding a particular thing, is given by C(n-1, r). This can be derived using the combination formula and simplifying the expression. The correct answer is option b) - C(n-1, r).
The number of selections of n different things taken r at a time can be calculated using the combination formula, denoted as "nCr" or "C(n,r)". The formula is given by:
C(n,r) = n! / (r! * (n-r)!)
Here, n! represents the factorial of n, which is the product of all positive integers less than or equal to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120.
Now, let's consider the scenario where we want to select r things from n different things, excluding a particular thing. In this case, we have (n-1) things to choose from as we are excluding one thing.
Explanation:
To solve this problem, we need to find the number of selections of (n-1) different things taken r at a time.
Using the combination formula, we can write this as:
C(n-1, r) = (n-1)! / (r! * ((n-1)-r)!)
Simplifying this expression, we get:
C(n-1, r) = (n-1)! / (r! * (n-r-1)!)
Now, let's compare this expression with the given options:
a) nC(n-1, r-1) = n! / ((r-1)! * (n-r)!)
b) C(n-1, r) = (n-1)! / (r! * (n-r-1)!)
c) rC(n-1, r) = r! * ((n-1)! / ((r-1)! * (n-r)!)
d) none of these
Comparing option b) with the derived expression, we can see that they are the same. Therefore, the correct answer is option b) - C(n-1, r).
Summary:
The number of selections of n different things taken r at a time, excluding a particular thing, is given by C(n-1, r). This can be derived using the combination formula and simplifying the expression. The correct answer is option b) - C(n-1, r).
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The number of selections of n different things taken r at a time which...
Option B is correct because it has mentioned that
we have to exclude a particul
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