In a certain test,there are n questions. In t | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $S_i$ be the number of students who answered at least $i$ questions wrongly. We are given $S_i = 2^{n-i}$ for $i = 1, 2, \dots, n$.The number

Vedclass · Accepted Answer

$11$

Vedclass · Answer

(B) Let $S_i$ be the number of students who answered at least $i$ questions wrongly. We are given $S_i = 2^{n-i}$ for $i = 1, 2, \dots, n$.The number of students who answered exactly $i$ questions wrongly is $N_i = S_i - S_{i+1}$ for $1 \le i The total number of wrong answers is given by the sum $\sum_{i=1}^{n} i \cdot N_i$.Total wrong answers $= 1(S_1 - S_2) + 2(S_2 - S_3) + 3(S_3 - S_4) + \dots + (n-1)(S_{n-1} - S_n) + n(S_n)$.Expanding this sum: $S_1 + S_2 + S_3 + \dots + S_n$.Substituting $S_i = 2^{n-i}$:Total wrong answers $= 2^{n-1} + 2^{n-2} + \dots + 2^0$.This is a geometric series with sum $\frac{2^n - 1}{2 - 1} = 2^n - 1$.Given $2^n - 1 = 2047$,we have $2^n = 2048 = 2^{11}$.Therefore,$n = 11$.

In a certain test,there are $n$ questions. In the test,$2^{n-i}$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, \dots, n$. If the total number of wrong answers given is $2047$,then $n$ is equal to

Similar Questions

If $11$ members of a committee sit at a round table so that the President and Secretary always sit together,then the number of arrangements is

The value of ${}^{50}{C_4} + \sum_{r = 1}^6 {^{56 - r}{C_3}}$ is

There are $6$ multiple choice questions in an examination. How many sequences of answers are possible,if the first three questions have $4$ choices each and the next three have $5$ choices each?

The number of arrangements of the letters of the word $SATAYPAUL$ such that no two $A$ are together and the middle letter is a consonant,is

The sum of all $4$ digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is

Vedclass Test Series

Exam Paper Generator

Online Exam Module