Home
Class 12
MATHS
The remainder when (sum(k=1)^(5) ""^(20)...

The remainder when `(sum_(k=1)^(5) ""^(20)C_(2k-1))^(6)` is divided by 11, is :

Text Solution

Verified by Experts

The correct Answer is:
3
We have `E = (underset(r=1)overset(5)sum.^(20)C_(2r-1))^(6)`
We know that `.^(20)C_(1)+.^(20)C_(3)+.^(20)C_(5)+.^(20)C_(7)+"......"+.^(20)C_(19)=2^(19)`
Now, `.^(20)C_(1)=.^(20)C_(19),.^(20)C_(3)=.^(20)C_(17)"....."`etc.
`:. 2(.^(20)C_(1)+.^(20)C_(3)+"...."+.^(20)C_(9))=2^(19)`
`rArr .^(20)C_(1)+.^(20)C_(3)+"....."+.^(20)C_(9)=2^(18)`
`rArr E = (2^(18))^(6) = 2^(108)`
`= 8(2^(5))^(21)`
`= 8(33-1)^(21)`
`= 8(33k-1)`
`= 8 xx 33k - 8`
`= 11(8xx3k-1)+3`
Therefore, the remainder when `(underset(r=1)overset(5)sum.^(20)C_(2r-1))^(6)` is divided by 11 is 3.
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.5|8 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.6|10 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.3|7 Videos

  • AREA

    CENGAGE PUBLICATION|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise For problems 3 and 4|2 Videos

Similar Questions

Explore conceptually related problems

Find the sum sum_(k=0)^(10).^(20)C_k .

Find the sum sum_(k=0)^(15).^(30)C_k .

sum_(k=1)^ook(1-1/n)^(k-1) =?

If A is a square matric of order 5 and 2A^(-1)=A^(T) , then the remainder when |"adj. (adj. (adj. A))"| is divided by 7 is

Remainder when 5^(40) is divided by 11.

Find the remainder when x^5+ 5x^4 + x^3 + 5x^2 + 2x + 11 is divided by x + 5

Let A={0,1,2,3,4,5}. If a_(1)binA, then an operation @ on A is defined by a@b=k where k is the least non-negative remainder when the sum (a+b) is divided by 6. Show that @ is a binary operation on A.

The value of cot(sum_(n=1)^(23)cot^-1(1+sum_(k=1)^n2k)) is

The value of cot {sum_(n=1)^(23)cot^(-1)""(1+sum_(k-1)^(n)2k)} is