Question

if nc4 = 210 then n=

Answer 1

Given,

nC4 = 210

To find,

The value of n

Solution,

We can easily solve this mathematical problem by using the following mathematical process.

The value of the given combination will be

= nC4

= (n!)/(4!) × (n-4)!

According to the data mentioned in the question,

(n!)/(4!) × (n-4)! = 210

{n × (n-1) × (n-2) × (n-3) × (n-4)!}/{(4!) × (n-4)!} = 210

n × (n-1) × (n-2) × (n-3) = 210 × 24

Now,

n, (n-1), (n-2),(n-3) are four consecutive numbers.

So, value of (210×24) is equal to the multiplication of four consecutive numbers.

Now, (210×24) can be rewritten as,

= 210×24

= 3×7×10×3×8

= 10×9×8×7

So,

n × (n-1) × (n-2) × (n-3) = 10×8×9×7

Implies that,

n = 10

Hence, the value of n is 10.

Answer 2

SOLUTION

GIVEN

$\sf{ {}^{n}C_4 = 210 }$

FORMULA TO BE IMPLEMENTED

$\displaystyle \sf{ {}^{n}C_r = \frac{n ! }{r! \times (n - r)! } }$

EVALUATION

$\sf{ {}^{n}C_4 = 210 }$

$\displaystyle \sf{ \implies {}^{n}C_4 = 10 \times 7 \times 3 }$

$\displaystyle \sf{ \implies {}^{n}C_4 = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 } }$

$\displaystyle \sf{ \implies {}^{n}C_4 = \frac{10 \times 9 \times 8 \times 7 \times 6! }{4! \times 6! } }$

$\displaystyle \sf{ \implies {}^{n}C_4 = \frac{10 ! }{4! \times 6! } }$

$\displaystyle \sf{ \implies {}^{n}C_4 = {}^{10}C_4 }$

$\displaystyle \sf{ \implies n = 10}$

FINAL ANSWER

Hence the required value of n = 10

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