Question

There are 20 questions in a question paper. If no two students solve the same combination of questions but solve equal number of questions then the maximum number of students who appeared in the examination is
$
  A.{\,^{20}}{C_9} \\
  B.{\,^{20}}{C_{11}} \\
  C.{\,^{20}}{C_{10}} \\
  D.\,{\text{none}}\,{\text{of}}\,{\text{these}} \\
$

Answer 1

Hint – In order to get this problem correct we need to use combinations and the fact that here in this case the number of student is maximum if the number of problems is half of the total number of problems according to general series of combination the largest term of all the terms are that which is half of the total number. Knowing this will solve your problem.

Complete step by step solution:
Let the number of problems solved by each student be x.
Then we need to maximize the selection $^{20}{C_x}$.
Now let us consider a series of combination so, that we can know which term is the largest of all the terms present in the series,
So we do,
$^n{C_1}{ + ^n}{C_2}{ + ^n}{C_3} + {..........^n}{C_{n - 1}}{ + ^n}{C_n}$.
The highest term in the above series will be $^n{C_{\dfrac{n}{2}}}$.
So, the largest term in the series of $^{20}{C_x}$ will be $^{20}{C_{\dfrac{{20}}{2}}}{ = ^{20}}{C_{10}}$
So, the maximum number of students will be $^{20}{C_{10}}$.
Hence, the correct option is C.

Note – When you get to solve such problems you need to know that whenever you need to do selection you need to use the combination and whenever you need to do arrangements you need to use the permutation. Then knowing that half the total number of selections is the largest in the series of combinations. Knowing this will solve your problem and will give you the right answer.