Question

Given that 5 answer sheets can be checked by any of 8 teachers. If the probability that all the 5 answer sheets are checked by exactly 3 teachers is $\tfrac{abc}{2^n}$ where abc be a 3‑digit number & n be the least natural number, then value of $(3a+2b+c-n)$ is equal to:

Answer 1

13

Answer 2

Total ways to assign 5 sheets to 8 teachers:

$$8^5=32768.$$

Favourable ways: exactly 3 distinct teachers check the sheets.

1. Choose 3 teachers: $\binom{8}{3}=56$.
2. Distribute 5 sheets onto these 3 teachers so each gets ≥1:

$$\text{Onto count} =3^5-3\cdot2^5+3\cdot1^5 =243-96+3=150.$$

Total favourable =$56\times150=8400$.
So probability =$\frac{8400}{32768}=\frac{525}{2048}=\frac{abc}{2^n}$ with $abc=525,\;2^n=2048\Rightarrow n=11$.
Thus $a=5,b=2,c=5,n=11$ and

$$3a+2b+c-n =3\cdot5+2\cdot2+5-11=15+4+5-11=13.$$