Question

In an engineering entrance examination, there are only Multiple select questions (more than
one option is correct). If a student selects any of the correct options, she will be awarded partial
marks in proportion to the number of correct options she selects. If she chooses any of the wrong
options, she will get zero marks for the entire question. In a particular question there are four
options of which the first three options are correct. In how many ways can the student get less
than half of the marks allotted to the question assuming the student selects atleast one option?​

Answer 1

Given :  If a student selects any of the correct options, she will be awarded partial marks in proportion to the number of correct options she selects.  

If she chooses any of the wrong options, she will get zero marks for the entire question.  

In a particular question, there are four options of which the first three options are correct.

To Find : In how many ways can the student get less than half of the marks allotted to the question assuming the student selects atleast one option

Solution:

Assume That Question is of 3X  Marks

Then 1 correct option = X mark

2 Correct option = 2X marks

3 Correct option = 3X

Marks can be 0 , X , 2X or 3X

Less than Half mark  = 3X/2 = 1.5X  mark

Hence 0 or X  marks

As students selects at least one option

Hence 0 Marks if wrong option (4th One is selected )

now in remaining 3 options , Either can be selected or not

Hence 2 x 2 x 2 = 8 ways

or ( in 3 option 0 , 1 , 2 or 3 options can be selected  ³C₀ + ³C₁ +³C₂ + ³C₃ = 1 + 3 + 3 + 1  = 8 ways)

Now X mark can be obtained by selecting any of the three correct options

hence 3 ways

So Total ways = 8 + 3 = 11

in 11 Ways   student get less than half of the marks allotted to the question assuming the student selects atleast one option

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