Home
Class 12
MATHS
Let S(1) = sum(j=1)^(10) j(j-1).""^(10)...

Let `S_(1) = sum_(j=1)^(10) j(j-1).""^(10)C_(j), S_(2) = sum_(j=1)^(10)j.""^(10)C_(j)`, and `S_(3) = sum_(j=1)^(10) j^(2).""^(10)C_(j)`.
Statement 1 : `S_(3) = 55 xx 2^(9)`.
Statement 2 : `S_(1) = 90 xx 2^(8)` and `S_(2) = 10 xx 2^(8)`.

A

(a) Statement 1 is false, statement 2 is true.

B

(b) Statement 1 is true, statement 2 is true, statement 2 is a correct explanation for statement 1.

C

(c) Statement 1 is true, statement 2 is true, statement 2 is not a correct explanation for statement 2.

D

(d) Statement 1 is true, statement 2 is false.

Text Solution

Verified by Experts

The correct Answer is:
B
`S_(1)=underset(j=1)overset(10)sumj(j-1)(10!)/(j(j-1)(j-2)!(10-j)!)`
`= 90underset(j=1)overset(10)sum(8!)/((j-2)!(8-(j-2)!)`
`=90underset(j=2)overset(10)sum.^(8)C_(j-2)=90xx2^(8)`
`S_(1) = underset(j=1)overset(10)sum(10!)/(j(j-1)!(9-(j-1))!)`
`= 10underset(j=1)overset(10)sum(9!)/((j-1)!(9-(j-1))!)`
`10underset( j=1)overset(10)sum.^(9)C_(j-1)= 10 xx 2^(9)`
`S_(3) = underset(j=1)overset(10)sum[j(j-1)+j] .^(10)C_(j)`
`= underset(j=1)overset(10)sumj(j-1).^(10)C_(j)+underset(j=1)overset(10)sum..^(10)C_(j)`
`= 90xx2^(8)+10xx2^(9) = 55 xx 2^(9)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer|4 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Comprehension|11 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Numerical|25 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_(j=0)^n( ^(4n+1)C_j+^(4n+1)C_(2n-j)) .

Evaluate : .^(20)C_(5)+sum_(j=2)^(5).^(25-j)C_(4)

The sum sum_(k=1)^(10)underset(i ne j ne k)underset(j=1)(sum^(10))sum_(i=1)^(10)1 is equal to

The sum sumsum_(0leilejle10) (""^(10)C_(j))(""^(j)C_(i-1)) is equal to

sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(a^(i+j+k)) is equal to (where |a| gt 1 )

sum_(i = 1)^n sum_(j = 1)^i sum_(k = 1)^j 1 is equal to

Find the following sum: sumsum_(i ne j) ""^(n)C_(i).""^(n)C_(j)

Calculate the momentum of a photon of frequency 5 xx 10^(13) Hz .Given, h = 6.6 xx 10^(-34)J.s and c = 3 xx 10^(8) m.s^(-1)

If A=([a_(i j)])_(nxxn) is such that ( a )_(i j)=bar (a_(j i)),AAi ,j and A^2=O , then Statement 1: Matrix A null matrix. Statement 2: |A|=0.

Let S_(1)=underset(0 le i lt j le 100)(sumsum)C_(i)C_(j) , S_(2)=underset(0 le j lt i le 100)(sumsum)C_(i)C_(j) and S_(3)=underset(0 le i = j le 100)(sumsum)C_(i)C_(j) where C_(r ) represents cofficient of x^(r ) in the binomial expansion of (1+x)^(100) If S_(1)+S_(2)+S_(3)=a^(b) where a , b in N , then the least value of (a+b) is