Question

The number of ways in which an examiner can assign \[30\] marks to \[8\] questions, giving not less than two marks to any question, is
A. \[{}^{30}{C_7}\]
B. \[{}^{21}{C_8}\]
C. \[{}^{21}{C_7}\]
D. \[{}^{30}{C_8}\]

Answer 1

Hint: In the given question, we need to find the number of ways in which an examiner can assign \[30\] marks to \[8\] questions, giving not less than two marks to any question. For this, we will assign the variables to marks assigned to the questions. Also, by using the given condition, we will find the desired result.

Formula used: The following formula used for solving the given question.
The formula of combination is given by, \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Here, \[n\] is the total number of things and \[r\] is the number of things that need to be selected from total things.

Complete step by step solution: We know that \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Here, \[n\] is the total number of things and \[r\] is the number of things that need to be selected from total things.
Now, let \[{x_i}\] be the marks assigned to the question.
Thus, we can say that \[{x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} + {x_8} = 30\]
Here, according to the given condition, we get
\[{x_i} \ge 2\]
 Here, \[i = 1,2,3,4,5,6,7,8\]
Also, suppose that \[y{}_i = {x_i} - 2\]
Here, \[i = 1,2,3,4,5,6,7,8\]
Thus, we can say that
\[{y_1} + {y_2} + {y_3} + {y_4} + {y_5} + {y_6} + {y_7} + {y_8} = ({x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} + {x_8}) - 8 \times 2\]
By simplifying, we get
\[{y_1} + {y_2} + {y_3} + {y_4} + {y_5} + {y_6} + {y_7} + {y_8} = 30 - 16\]
\[{y_1} + {y_2} + {y_3} + {y_4} + {y_5} + {y_6} + {y_7} + {y_8} = 14\]
So, the total number of solutions to the above equations is given by
\[{}^{14 + {\rm{8}} - {\rm{1}}}{C_{8 - 1}} = {}^{21}{C_7}\]
Hence, the number of ways in which an examiner can assign \[30\] marks to \[8\] questions, giving not less than two marks to any question, is \[{}^{21}{C_7}\].

Thus, Option (C) is correct.

Note: Many students make mistakes in calculation as well as writing the equation according to the given condition. This is the only way, through which we can solve the example in the simplest way. Also, it is essential to find correct equation such as \[{y_1} + {y_2} + {y_3} + {y_4} + {y_5} + {y_6} + {y_7} + {y_8} = 14\] to get the desired result.