Question

A person predicts the outcome of $20$ cricket matches of his home team. Each match can result either in a win, loss or tie for the home team. The total number of ways in which he can make the predictions such that exactly $10$ predictions are correct, are equal to

• A. ${}^{20}{C}_{10}\cdot 2^{10}$
• B. ${}^{20}{C}_{10}\cdot 3^{20}$
• C. ${}^{20}{C}_{10}\cdot 3^{10}$
• D. ${}^{20}{C}_{10}\cdot 2^{20}$

Answer 1

Answer: (A)

In order to predict exactly $10$ correct predictions,
$\Rightarrow$ we have to select $10$ matches out of $20$ in ${}^{20}{C}_{10}$ ways
$\Rightarrow$ ways of making a correct prediction for each match is $1$ , so for $10$ matches it is equal to $1^{10}=1.$
$\Rightarrow$ Remaining $10$ predictions must be wrong so for each incorrect prediction we have $2$ choices out of $3,$ this can be done in $2^{10}$ ways.
Hence, the required number of ways ${=}^{20}{C}_{10}\left(1\right)\cdot 2^{10}{=}^{20}{C}_{10}\times 2^{10}.$