Question

State whether the given statements are true or false.
Statement 1: Number of ways in which India can win the series of 11 matches in \[{{2}^{10}}\]. (if no match is drawn).
Statement 2: For each match there are two possibilities, either India wins or loses.

Answer 1

Hint: We will first start by checking the first statement whether this statement is true or not. We will use the fundamental principle of counting to find the ways in which India can win a series of 11 matches.

Complete step-by-step answer:
Now, in statement 1 we have been given the number of ways in which India can win the series of 11 matches is \[{{2}^{10}}\].
To find the ways in which India can win series we have to note that India can win a series if it won 6 or more matches out of 7. So, the total ways this can happen is,
$S={}^{11}{{C}_{6}}+{}^{11}{{C}_{7}}+{}^{11}{{C}_{8}}+{}^{11}{{C}_{9}}+{}^{11}{{C}_{10}}+{}^{11}{{C}_{11}}$
Now, we know ${}^{n}{{C}_{r}}={}^{n}{{C}_{n-r}}$. Therefore, we can write,
$S={}^{11}{{C}_{5}}+{}^{11}{{C}_{4}}+{}^{11}{{C}_{3}}+{}^{11}{{C}_{2}}+{}^{11}{{C}_{1}}+{}^{11}{{C}_{0}}$
Now, adding both we have,
$2S={}^{11}{{C}_{0}}+{}^{11}{{C}_{1}}+............+{}^{11}{{C}_{11}}$
Now, we that the sum of series,
${}^{n}{{C}_{0}}+{}^{n}{{C}_{1}}+............+{}^{n}{{C}_{n}}={{2}^{n}}$
Therefore, we have,
$\begin{align}
  & 2S={{2}^{11}} \\
 & S={{2}^{10}} \\
\end{align}$
Therefore, the first statement is true.
Now, in the second statement we have given that for each match there are two possibilities, either India wins or loses.
This statement is also true because it has been given in statement 1 that no match is drawn.

Note: It is important to note that we have used a fact while solving the problem that the sum ${}^{n}{{C}_{0}}+{}^{n}{{C}_{1}}+............+{}^{n}{{C}_{n}}={{2}^{n}}$. Also, it has to be noted that how we have used the property ${}^{n}{{C}_{r}}={}^{n}{{C}_{n-r}}$ to find the value of S.