Let S ₁= _ j =1 ¹⁰ j ( j -1)¹⁰ C _ j , S ₂= _ j =1 ¹⁰ j ¹⁰ C _ j and S ₃= _ j =1 ¹⁰ j ^2 ¹⁰ C _ j . Statement-1: S ₃=55 2^9 Statement-2: S₁=90 2^8 and S₂=10 2^8 .

Question

Quantrex Academy · Accepted Answer

Correct answer: B. Statement-1 is true, Statement-2 is false

S₁= _ j=1 ¹⁰ j(j-1) 10 ! j(j-1)(j-2) !(10-j) ! =90 _ j=2 ¹⁰ 8 ! (j-2) !(8-(j-2)) ! =90 2^8 . S₂= _ j=1 ¹⁰ j 10 ! j(j-1) !(9-(j-1)) ! =10 _ j=1 ¹⁰ 9 ! (j-1) !(9-(j-1)) ! =10 2^9 . S₃= _ j=1 ¹⁰[j(j-1)+j] 10 ! j !(10-j) ! = _ j=1 ¹⁰ j(j-1)¹⁰ C_j= _ j=1 ¹⁰ j¹⁰ C_j=90 2^8+10 2^9=90 2^8+20 2^8=110 2^8=55 2^9 .

Let S ₁= _ j =1 ¹⁰ j ( j -1)¹⁰ C _ j , S ₂= _ j =1 ¹⁰ j ¹⁰ C _ j and S ₃= _ j =1 ¹⁰ j ^2 ¹⁰ C _ j . Statement-1: S ₃=55 2^9 Statement-2: S₁=90 2^8 and S₂=10 2^8 .

Options

Correct answer

Step-by-step solution

More from Binomial Theorem