An examiner can assign 30 marks to 8 question | vedclass

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

Question

(A) Let $x_i$ be the marks assigned to the $i$-th question,where $i = 1, 2, \dots, 8$.We are given that $x_1 + x_2 + \dots + x_8 = 30$ and $x_i \ge 2$

Vedclass · Accepted Answer

$^{21}C_7$

Vedclass · Answer

(A) Let $x_i$ be the marks assigned to the $i$-th question,where $i = 1, 2, \dots, 8$.We are given that $x_1 + x_2 + \dots + x_8 = 30$ and $x_i \ge 2$ for all $i$.Let $y_i = x_i - 2$,so $y_i \ge 0$.Substituting $x_i = y_i + 2$ into the equation:$(y_1 + 2) + (y_2 + 2) + \dots + (y_8 + 2) = 30$$y_1 + y_2 + \dots + y_8 + 16 = 30$$y_1 + y_2 + \dots + y_8 = 14$The number of non-negative integer solutions is given by the formula $^{n+r-1}C_{r-1}$,where $n = 14$ and $r = 8$.Number of ways = $^{14+8-1}C_{8-1} = ^{21}C_7$.

An examiner can assign $30$ marks to $8$ questions. If he does not assign less than $2$ marks to any question,in how many ways can he assign the marks?

Similar Questions

The number of ordered triplets of positive integers which are solutions of the equation $x + y + z = 100$ is

If $^{n}C_{9} = ^{n}C_{8}$,find $^{n}C_{17}$.

Compute $\frac{8!}{6! \times 2!}$

In a touring cricket team,there are $16$ players in all,including $5$ bowlers and $2$ wicket-keepers. How many teams of $11$ players can be chosen from these,such that the team includes exactly $3$ bowlers and $1$ wicket-keeper?

$A$ committee of $3$ persons is to be constituted from a group of $2$ men and $3$ women. In how many ways can this be done? How many of these committees would consist of $1$ man and $2$ women?

Vedclass Test Series

Exam Paper Generator

Online Exam Module