16. For what values of n is n-1C3 =
4(nC3)?
A. 6
B. 5
C. 4
D. 3
To solve this problem, we need to express the combinations in terms of factorials.
The formula for nCr is given by nCr = n! / (r!(n-r)!).
Using this formula, we can express n-1C3 as (n-1)! / (3!((n-1)-3)!), which simplifies to (n-1)! / 3!(n-4)! = (n-1)(n-2)(n-3) / 3!.
Similarly, nC3 can be expressed as n! / (3!(n-3)!), which simplifies to n(n-1)(n-2) / 3!.
We are given that n-1C3 = 4(nC3), so we can set up the equation:
(n-1)(n-2)(n-3) / 3! = 4(n)(n-1)(n-2) / 3!
Simplifying, we have:
(n-3) = 4n
n - 3 = 4n
-3 = 3n
n = -1
However, since we are looking for values of n that make the equation valid, we must check if n = -1 satisfies the equation or not.
Plugging in n = -1, we have:
(-1 - 3) = 4(-1)
-4 = -4
Since the equation is true, n = -1 is a valid solution.
Therefore, the answer is n = -1.
The correct option is not given in the answer choices.
To find the values of n that satisfy the equation n-1C3 = 4(nC3), we can use the formula for combinations.
First, let's expand the terms using the combination formula:
(n-1)C3 = ((n-1)!)/((3!)(n-1-3)!) = ((n-1)!)/(3!(n-4)!)
4(nC3) = 4(n!)/((3!)(n-3)!)
Now we can equate the two expressions:
((n-1)!)/(3!(n-4)!) = 4(n!)/((3!)(n-3)!)
Next, we can simplify the equation by canceling out common factors:
(n-1)!/(n-4)! = 4(n!)/(n-3)!
Now, let's focus on canceling out the factorials. We can notice that the factorials in both fractions can be canceled out up to the (n-4) term:
(n-1)(n-2)(n-3)(n-4)!/(n-4)! = 4n(n-1)(n-2)(n-3)/(n-3)(n-4)!
(n-1)(n-2)(n-3) = 4n(n-1)(n-2)
Now, let's simplify the equation further:
(n-1)(n-2)(n-3) - 4n(n-1)(n-2) = 0
Now, we can start canceling out the common factors again:
(n-1)(n-2)[(n-3)-4n] = 0
(n-1)(n-2)(-3n-3) = 0
Now, we have three factors that could be equal to zero:
1. (n-1) = 0
2. (n-2) = 0
3. (-3n-3) = 0
Solving these equations, we get:
1. n-1 = 0
n = 1
2. n-2 = 0
n = 2
3. -3n-3 = 0
-3n = 3
n = -1
Therefore, the values of n that satisfy the equation n-1C3 = 4(nC3) are n = 1, n = 2, and n = -1.
Answer: There is no option provided in the given question that includes all three of the possible values for n.