The number of ways in which an examiner can a | vedclass

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

Question

(C) Let the marks assigned to the $8$ questions be $x_1, x_2, \ldots, x_8$.
We are given that:
$x_1 + x_2 + \cdots + x_8 = 30$, where $x_i \ge 2$ fo

Vedclass · Accepted Answer

$^{21}C_7$

Vedclass · Answer

(C) Let the marks assigned to the $8$ questions be $x_1, x_2, \ldots, x_8$.
We are given that:
$x_1 + x_2 + \cdots + x_8 = 30$, where $x_i \ge 2$ for each $i \in \{1,2,\ldots,8\}$.
Let $x_i = y_i + 2$, where $y_i \ge 0$.
Substituting:
$(y_1 + 2) + (y_2 + 2) + \cdots + (y_8 + 2) = 30$
$y_1 + y_2 + \cdots + y_8 + 16 = 30$
$y_1 + y_2 + \cdots + y_8 = 14$
The number of non-negative integer solutions is given by:
$\binom{n+r-1}{r-1}$, where $n=14$ and $r=8$.
Number of ways:
$\binom{14+8-1}{8-1} = \binom{21}{7} = {}^{21}C_7$

The number of ways in which an examiner can assign $30$ marks to $8$ questions,giving not less than $2$ marks to any question,is

Similar Questions

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$,$Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

Assuming the balls to be identical except for the difference in colours,the number of ways in which one or more balls can be selected from $10$ white,$9$ green,and $7$ black balls is:

The number of words formed from the letters of the word $RAJASTHAN$ by taking all the letters at a time in which vowels are alternate is:

The sum of all three-digit numbers,formed using non-zero digits,where all the digits are perfect squares of a natural number,is:

Out of $13$ cricket players,$4$ are bowlers. In how many ways can a cricket team of $11$ players be formed such that at least $2$ bowlers are included?

Vedclass Test Series

Exam Paper Generator

Online Exam Module