Question

In how many ways 30 marks can be allotted to 8 questions if each question carries at least 2 marks?

Answer 1

Hint: There are total 8 questions and each question carries at least 2 marks. So, $ 8 \times 2 = 16 $ marks are allotted and we have to find the number of ways to allot the remaining $ 30 - 16 = 14 $ marks to the 8 questions. To find these number of ways, we are using the formula of combination $ {}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $ .

Complete step by step solution:
Given data:
Number of questions = 8
Total Marks = 30
We start with 30 marks in the bank to allot.
 Since, all the questions must carry at least 2 marks, we take $ 2 \times 8 = 16 $ marks out of the 30 marks and allot them equally.
Now, each question has 2 marks as of now and the bank is left with $ 30 - 16 = 14 $ marks.
Now, we need to find the number of ways to allot these 14 marks among 8 questions.
At first, it might seem very hard, but there is a trick to make it very easy.
Now to find the total number of ways in which we can allot these 14 marks to 8 questions, we are going to use the concept of combination.
 $ {}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $
The number to the left of C is equal to the number of marks we are splitting plus the number of splitters. The number of splitters is always one less than the question. The number on the right of C is the number of marks.
Therefore, we calculate
\[
   \Rightarrow {}^{14 + 8 - 1}{C_{14}} = \dfrac{{21!}}{{14!\left( {21 - 14} \right)!}} \\
   \Rightarrow {}^{21}{C_{14}} = \dfrac{{21!}}{{14!7!}} \;
 \]
On further simplifying the above equation, we get
\[ \Rightarrow {}^{21}{C_{14}} = 116280\]
Therefore, there are 116280 ways to allot 30 marks to 8 questions if each question carries at least 2 marks.
So, the correct answer is “116280 ”.

Note: The difference between permutation and combination is ordering. Permutation cares about the order of elements, whereas combination does not. For example say your locker combo is 1234 and if you enter 4321, the locker won’t open due to different ordering.