Passing GCSE O Level Examinations

For any possibility of passing and failing, we can reverse all the grades on the subjects. Exactly one of these two possibilities gives Robbie a pass (the other one fails). So exactly half of all possibilities of grades give him a pass. There are $2^9 = 512$ possibilities (each subject has two possibilities, pass or fail, and there are nine subjects), so exactly half of these, or $\frac{1}{2} \cdot 512 = \boxed{256}$ , gives Robbie a pass.

To pass his exams he need to pass 5 subjects or more out of 9. Hence the number of ways N can be given by: N= 9C5+9C6+9C7+9C8+9C9 N= 256 And we add all the combinations because these things cannot happen together. He can either pass 6 or 7 subjects, for example, but he cannot pass 6 and 7 subjects. Cheers

We can consider him passing in either 9 or 8 or 7 or 6 or 5 subjects. If we take the combination of this, it automatically means that he had failed the remaining.

So 9C9 +9C8 + 9C7 + 9C6 + 9C5

Since 5 is mean for 0 and 9, The above sun is equal to ( 2^9 )÷2 Which is 256..

if he passes the examination , that means he has passed all the subjects or 8 subjects, or 7 or 6 or 5. so- passing all subjects= 1 possibility. now, there are 9 combinations.combination of these taking 1 at time (1 refers to failing in 1 subject) = 9 = 9 more possibilities .same can be done for 2 subjects at a time, to get = 36 possibilities ,for 3 at a time = 84 possibilities ,for 4 at a time = 126 possibilities .therefore total possibilities = 1+9+36+84+126=256

{note - combination of n distinct elements taking r at a time = n! divided by [r! * (n-1)!] }

It is simple to pass you may Pass all courses. 1 Pass 8 courses there are 9 ways one fir each course failed Pass 7 courses this is 9x8/2 = 36 Pass 6 courses this is 9x8x7/(3x2) = 84 Pass 5 courses this is 9x8x7x6/(4x3x2) = 126 Total. 1+ 9+36+84+126 = 256

To pass he may pass in 5,6,7,8 or 9 subjects. This can be done in $\binom{9}{9}+\binom{9}{8}+\binom{9}{7}+\binom{9}{6}+\binom{9}{5}=256ways$