Question

Let x = (10c1)2 + 2(10c2)2 + 3(10c3)2 + . + 10(10c10)2 , where 10cr , r {1, 2, .. , 10} denote binomial coefficients. Then the value of 1/1430 x is ____ .

Answer 1

Given: x = (10c1)² + 2(10c2)² + 3(10c3)² +  .......  + 10(10c10)²

To find: The value of x/1430 ?

Solution:

• Now we have given the binomial coefficient as :

           x = (10c1)² + 2(10c2)² + 3(10c3)² +  .......  + 10(10c10)²

• It can be written as:

               n

         x = ∑ r.(nCr)² ................where n = 10

             r = 0

               n

         x = n. ∑ (nCr) . (n-1 C r-1)

             r = 0

                    n

         x = n.  ∑ (n C n-r) . (n-1 C r-1)

                 r = 0

         x = n( 2n-1 C r-1)

         x = 10 ( 19 C 9)

• Now putting this value in x/1430, we get:

• 10 (19 C 9) / 1430 = (19 C 9) / 143

• After calculation the answer comes out to be: 646.

Answer:

           So the value of x/1430 is 646