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  4. 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

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Q. 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

NTA AbhyasNTA Abhyas 2020Permutations and Combinations

Solution:

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 $=^{20}C_{10}\left(1\right)\cdot 2^{10}=^{20}C_{10}\times 2^{10}.$