Q

# I have a doubt, kindly clarify. There are three papers of 100 marks each in an examination. Then the number of ways a student gets 150 marks such that he gets at least60% in two papers.

There are three papers of 100 marks each in an examination. Then the number of ways a student gets 150 marks such that he gets at least 60% in two papers.

• Option 1)

3C2 x 32C2

• Option 2)

4C3 x 32C2

• Option 3)

4Cx 36C2

• Option 4)

4Cx 32C3

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As learnt in

Theorem of Combinations -

The number of combinations of n distinct objects taken r at a time when any object may be repeated any number of times is $^{n+r-1}c_{r}$.

- wherein

Coefficient of $x^{r}$ in $(1-x)^{-n}$.

150 marks

$x+y+z=150$

$x \geqslant60$

$y \geqslant60$

$z \geqslant 0$

Thus, $x'+y'+z'=30$

We get

$^{30+3-1}C_{3-1}=^{32}C_{2} \:$  solutions

and the two papers can be chosen in $^{3}C_{2}$ ways.

Option 1)

3C2 x 32C2

This option is correct.

Option 2)

4C3 x 32C2

This option is incorrect.

Option 3)

4Cx 36C2

This option is incorrect.

Option 4)

4Cx 32C3

This option is incorrect.

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