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Chapter 5: Problem 406

If \(\mathrm{n}={ }^{\mathrm{m}} \mathrm{C}_{2}\) then \({ }^{\mathrm{n}} \mathrm{C}_{2}\) equal to. (a) \({ }^{\mathrm{m}+1} \mathrm{C}_{4}\) (b) \({ }^{\mathrm{m}-1} \mathrm{C}_{4}\) (c) \({ }^{\mathrm{m}+2} \mathrm{C}_{4}\) (d) None of these

Short Answer

Expert verified
The short answer is: (d) None of these.

Step by step solution

01

Write the given equation using the combination formula

First, we rewrite the given equation \(n = ^mC_2\) using the combination formula: \(n = \frac{m!}{2!(m-2)!}\)
02

Solve for m! to express n in terms of m

Next, we need to solve for m! in the above equation to get n in terms of m: \(m! = n\cdot2!(m-2)!\)
03

Substitute m! to express n in terms of m

We substitute m! in our original equation to express n in terms of m: \(n = \frac{n\cdot2!(m-2)!}{2!(m-2)!}\)
04

Simplify the equation

Next, we need to simplify the equation to express n in terms of m: \(n = m(m-1)\)
05

Write the expression for ^nC_2 in terms of m

Now, we need to substitute \(n = m(m-1)\) into the basic combination formula, \(^nC_2 = \frac{n!}{2!(n-2)!}\): \(^nC_2 = \frac{[(m(m-1))]!}{2![(m(m-1))-2]!}\)
06

Compare options with the given expression to find the correct answer

Now we can compare our derived expression for \(^nC_2\) with the given options: (a) \(^mC_{m+1}\) (b) \(^mC_{m-1}\) (c) \(^mC_{m+2}\) (d) None of these. None of the given options match our derived expression for \(^nC_2\), so the correct answer is: (d) None of these.

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