Question 14 Marks
Ashish is writing examination. He is reading question paper during reading time. He reads instructions carefully. While reading instructions, he observed that the question paper consists of 15 questions divided in to two parts part I containing 8 questions and part II containing 7 questions.
i. If Ashish is required to attempt 8 questions in all selecting at least 3 from each part, then in how many ways can he select these questions (1)
ii. If Ashish is required to attempt 8 questions in all selecting 3 from I part, then in how many ways can he select these questions (1)
iii. If Ashish is required to attempt 8 questions in all selecting 4 from part I and 4 from part II, then in how many ways can he select these questions (2)
OR
If Ashish is required to attempt 8 questions in all selecting 6 from one section and remaining from another section, then in how many ways can he select these questions (2)
i. If Ashish is required to attempt 8 questions in all selecting at least 3 from each part, then in how many ways can he select these questions (1)
ii. If Ashish is required to attempt 8 questions in all selecting 3 from I part, then in how many ways can he select these questions (1)
iii. If Ashish is required to attempt 8 questions in all selecting 4 from part I and 4 from part II, then in how many ways can he select these questions (2)
OR
If Ashish is required to attempt 8 questions in all selecting 6 from one section and remaining from another section, then in how many ways can he select these questions (2)
Answer
View full question & answer→i.Since, at least 3 questions from each part have to be selected
So number of ways are
3 questions from part I and 5 questions from part II can be selected in $n^8 C_3 \times{ }^7 C_5$ ways
4 questions from part I and 4 questions from part II can be selected in ${ }^8 C_4 \times{ }^7 C_4$ ways
5 questions from part I and 3 questions from part II can be selected in ${ }^8 C_5 \times{ }^7 C_3$ ways
So required number of ways are
${ }^8 C_3 \times{ }^7 C_5+{ }^8 C_4 \times{ }^7 C_4+{ }^8 C_5 \times{ }^7 C_3$
$\begin{array}{l}
\Rightarrow \frac{8!}{5!\times 3!} \times \frac{7!}{5!\times 2!}+\frac{8!}{4!\times 4!} \times \frac{7!}{4!\times 3!}+\frac{8!}{5!\times 3!} \times \frac{7!}{4!\times 3!} \\
\Rightarrow \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \times \frac{7 \times 6}{2 \times 1}+\frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1}+\frac{8 \times 7 \times 6}{3 \times 2 \times 1} \times \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} \\
\Rightarrow 56 \times 21+70 \times 35+56 \times 35 \\
\Rightarrow 1176+2450+1960 \\
\Rightarrow 5586\end{array}$
ii. Ashish is selecting 3 questions from part I so he has to select remaining 5 questions from part II The number of ways of selection is
3 questions from part I and 5 questions from part II can be selected in ${ }^8 C_3 \times{ }^7 C_5$ ways
$\begin{array}{l}\Rightarrow{ }^8 C_3 \times{ }^7 C_5 \\ \Rightarrow \frac{8!}{5!\times 31} \times \frac{7!}{5!\times 2!} \\ \Rightarrow \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \times \frac{7 \times 6}{2 \times 1} \\ \Rightarrow 56 \times 21 \\ \Rightarrow 1176\end{array}$
iii. 4 questions from part I and 4 questions from part II can be selected
$\begin{array}{l}{ }^8 C_4 \times{ }^7 C_4 \\ \Rightarrow \frac{8!}{4 \times 4!} \times \frac{7}{4!3!} \\ \Rightarrow \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1} \\ \Rightarrow 70 \times 35 \\ \Rightarrow 2450\end{array}$
OR
6 questions from part I and 2 questions from part II can be selected or 2 questions from part I and 6 questions from part II can be selected
$\begin{array}{l}\Rightarrow{ }^8 C_6 \times{ }^7 C_2+{ }^8 C_2 \times{ }^7 C_6 \\ \Rightarrow \frac{8!}{6!\times 2!} \times \frac{7!}{2!\times 5!}+\frac{8!}{6!\times 2!} \times \frac{7!}{1!\times 6!}\end{array}$
$\begin{array}{l}\Rightarrow \frac{8 \times 7}{2 \times 1} \times \frac{7 \times 6}{2 \times 1}+\frac{8 \times 7}{2 \times 1} \times 7 \\ \Rightarrow 28 \times 21+28 \times 7 \\ \Rightarrow 588+196=784\end{array}$
| Part I | Part II |
| 3 | 5 |
| 4 | 4 |
| 3 | 5 |
3 questions from part I and 5 questions from part II can be selected in $n^8 C_3 \times{ }^7 C_5$ ways
4 questions from part I and 4 questions from part II can be selected in ${ }^8 C_4 \times{ }^7 C_4$ ways
5 questions from part I and 3 questions from part II can be selected in ${ }^8 C_5 \times{ }^7 C_3$ ways
So required number of ways are
${ }^8 C_3 \times{ }^7 C_5+{ }^8 C_4 \times{ }^7 C_4+{ }^8 C_5 \times{ }^7 C_3$
$\begin{array}{l}
\Rightarrow \frac{8!}{5!\times 3!} \times \frac{7!}{5!\times 2!}+\frac{8!}{4!\times 4!} \times \frac{7!}{4!\times 3!}+\frac{8!}{5!\times 3!} \times \frac{7!}{4!\times 3!} \\
\Rightarrow \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \times \frac{7 \times 6}{2 \times 1}+\frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1}+\frac{8 \times 7 \times 6}{3 \times 2 \times 1} \times \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} \\
\Rightarrow 56 \times 21+70 \times 35+56 \times 35 \\
\Rightarrow 1176+2450+1960 \\
\Rightarrow 5586\end{array}$
ii. Ashish is selecting 3 questions from part I so he has to select remaining 5 questions from part II The number of ways of selection is
3 questions from part I and 5 questions from part II can be selected in ${ }^8 C_3 \times{ }^7 C_5$ ways
$\begin{array}{l}\Rightarrow{ }^8 C_3 \times{ }^7 C_5 \\ \Rightarrow \frac{8!}{5!\times 31} \times \frac{7!}{5!\times 2!} \\ \Rightarrow \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \times \frac{7 \times 6}{2 \times 1} \\ \Rightarrow 56 \times 21 \\ \Rightarrow 1176\end{array}$
iii. 4 questions from part I and 4 questions from part II can be selected
$\begin{array}{l}{ }^8 C_4 \times{ }^7 C_4 \\ \Rightarrow \frac{8!}{4 \times 4!} \times \frac{7}{4!3!} \\ \Rightarrow \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1} \\ \Rightarrow 70 \times 35 \\ \Rightarrow 2450\end{array}$
OR
6 questions from part I and 2 questions from part II can be selected or 2 questions from part I and 6 questions from part II can be selected
$\begin{array}{l}\Rightarrow{ }^8 C_6 \times{ }^7 C_2+{ }^8 C_2 \times{ }^7 C_6 \\ \Rightarrow \frac{8!}{6!\times 2!} \times \frac{7!}{2!\times 5!}+\frac{8!}{6!\times 2!} \times \frac{7!}{1!\times 6!}\end{array}$
$\begin{array}{l}\Rightarrow \frac{8 \times 7}{2 \times 1} \times \frac{7 \times 6}{2 \times 1}+\frac{8 \times 7}{2 \times 1} \times 7 \\ \Rightarrow 28 \times 21+28 \times 7 \\ \Rightarrow 588+196=784\end{array}$
