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If $\mathrm{ n C r: n C r+1=1: 4}$ and $\mathrm{ n C r+1: n C r+2=4: 5}$ , determine the values of n and r.
Option: 1
25,000

Option: 2
60,000

Option: 3
10,000

Option: 4
40,000

Answers (1)

best_answer

If $\mathrm{n C r: n C r+1=1: 4}$ and , determine the values of n and r.

Let's solve the given equations to find the values of n and r :

From the first ratio, $\mathrm{n C r: n C r+1=1: 4}$ , we can write:

$\mathrm{ n C r+1=4 * n C r }$

From the second ratio, $\mathrm{n C r+1: n C r+2=4: 5}$ , we can write:

$\mathrm{ n C r+2=5 * n C r+1 }$

Now, we can solve these equations to find the values of n and r.

Substituting the value of $\mathrm{n C r+1}$ from the first equation into the second equation, we have:

$\mathrm{ n C r+2=5 *(4 * n C r) }$

Simplifying, we get:

$\mathrm{ n C r+2=20 * n C r }$

Dividing both sides by $\mathrm{n C r}$ , we have:

$\mathrm{ n C r+2 / n C r=20 }$
Using the property of binomial coefficients, we have:

$\mathrm{ (n+1) / r=20 }$
Simplifying further, we get:

$\mathrm{ n+1=20 r }$

From the first equation, $\mathrm{n C r+1=4 * n C r}$ , we have:

$\mathrm{ n C r+1=4 * n C r }$

Using the property of binomial coefficients, we have:

$\mathrm{ (n+1) /(r+1)=4 }$

Substituting $\mathrm{ n+1=20 r}$ into the above equation, we get:

$\mathrm{ 20 r /(r+1)=4 }$
Cross-multiplying, we have:

$\mathrm{ 20 r=4 r+4 }$

Subtracting 4 r from both sides, we have:

$\mathrm{ 16 r=4 }$

Dividing both sides by 16 , we get:

$\mathrm{ r=1 / 4 }$

Substituting the value of r into $\mathrm{n+1=20 r}$ , we have:

$\mathrm{ n+1=20 *(1 / 4) }$
Simplifying, we get:

$\mathrm{ n+1=5 }$

Subtracting 1 from both sides, we have:

$\mathrm{n=4}$ Therefore, the values of $\mathrm{n}$ and $\mathrm{r}$ that satisfy the given ratios are:

$\mathrm{ n=4 \text { and } r=1 / 4 }$
Hence, the solution for the values of n and r in the given equations is $\mathrm{n=4 \, \, and \, \, r=1 / 4.}$

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