mathematics-class-11-unit-08-chapter-03-permutation-combination-L-3-5-lq5e9wj1jh8

### <Success>Fundamental principles of counting</Success>
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<p><strong>1. The Addition Principle:</strong></p>
<p>Let these be $m$ ways for an event $A$ to occur and let these be $n$ ways for another event $B$ to occur. If all ways are distinct, thess the number of ways for $A$ or $B$ to occur is $(m+n)$.</p>
<p><strong>Remark:</strong> If we use the language of sets, then #(A) denotes the number of elements in set $A$ (cardinality of set $A$ ). Then the addition principle states, that if $A$ and $B$ are disjoint sets, $\#(A)=m, \ \ \#(B)=n$,</p>
<p>then $\# (A \cup B)= m+n$</p>
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<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-08-chapter-03-permutation-combination-L-3-5-lq5e9wj1jh8-7a.jpg"></p>
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### <Success>Example 1</Success>
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<p>From Delhi bo Mumbai, one can travel by air ( 8 flights), train ( 12 trains). and land (long distance bus service or car).
In how many ways one can travel from Delhi to Mumbai?</p>
<p><strong>Solution:</strong></p>
<p>$E_1 \rightarrow$ travelling by air, $\# (E_1)=8$</p>
<p>$E_2 \rightarrow$ Travelling by 1 train, $\# (E_2)=12$</p>
<p>$E_3 \rightarrow$ travelling by land $\# (E_3)=2$</p>
<p>So the total number of ways of travelling from Delhi to Mumbai is</p>
<p>$\# (E_1 \cup E_2 \cup E_3)  = \#(E_1)+ \# (E_2)+ \#(E_3) $</p>
<p>$ =8+12+2=22$</p>
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### <Success>Solution</Success>
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<p>$\# \left(E_1\right)=5^2=25 $</p>
<p>$\# \left(E_2\right)=(5-1)^2=16 $</p>
<p>$\# \left(E_3\right)=(5-2)^2=9 $</p>
<p>$\# \left(E_4\right)=(5-3)^2=4 $</p>
<p>$\# \left(E_5\right)=(5-4)^2=1$</p>
<p>So the total number of squares<br>
$=1+4+9+16+25 =55$</p>
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<p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/1a55786f-564e-47e9-7960-bd13a033ac00/public"></p>
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<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-08-chapter-03-permutation-combination-L-3-5-lq5e9wj1jh8-9a.jpg"></p>
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### <Success>Multiplication Principle</Success>
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<p>If there are $m$ ways of an event $A$ to occur and these are $n$ ways for an event $B$ to occur, then the total number of ways for occurrence of event $A$ followed by event $B$ are $\mathrm{mn}$.</p>
<p><strong>Proof:</strong>  Using the terminology of set theory.</p>
<p>let $A=\{a_1, \ldots, a_m \}$ and $B= \{b_1, \ldots, b_n\}$.</p>
<p>Then the possible number of ways of occurrence of event $A$ followed by event $B$ are</p>
<table>
<thead>
<tr>
<th>$(a_1, b_1)$</th>
<th>$(a_1, b_2)$</th>
<th>$\cdots$</th>
<th>$(a_1, b_n)$</th>
</tr>
</thead>
<tbody>
<tr>
<td>$(a_2, b_1)$</td>
<td>$(a_2, b_2)$</td>
<td>$\cdots$</td>
<td>$(a_2, b_n)$</td>
</tr>
<tr>
<td>$\vdots$</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>$(a_m, b_1)$</td>
<td>$(a_m, b_2)$</td>
<td>$\cdots$</td>
<td>$(a_m, b_n)$</td>
</tr>
</tbody>
</table>
<p>So the total number of ways is $m n$.</p>
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<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-08-chapter-03-permutation-combination-L-3-5-lq5e9wj1jh8-10a.jpg"></p>
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### <Success>General Multiplication Principle</Success>
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<p>Let these be $m_1$ ways for an event $E_1$ to occur, $m_2$ ways for event $E_2$ to occur, …, $m_k$ ways for event $E_k$ to occur. Then the total number of ways in which events $E_1, E_2, \ldots$ $E_k$ occur in this order is $m_1 m_2 \ldots m_k$.</p>
<p><strong>Remark:</strong>   In the multiplication principle, the order in which events occur does not make any difference since multiplication is commutative and also because the cardinality of cartesian product of sets does not depend upon the order in which sets are takes in a Cartesian product.</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-08-chapter-03-permutation-combination-L-3-5-lq5e9wj1jh8-11.jpg">
<img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/16d4dadc-cdca-4aff-6851-ae38cbbf8000/public">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-08-chapter-03-permutation-combination-L-3-5-lq5e9wj1jh8-11a.jpg"></p>
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### <Success>Example 1</Success>
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<p>In a class of 50 students, 20 are majoring in Physics, 20 are majoring in chemistry and 10 are majoring in Mathematics. In how many ways we can choose a representative from each group?</p>
<p><strong>Solution:</strong></p>
<p>$E_1 \rightarrow$ choosing representative from students majoring in Physics</p>
<p>$E_2 \rightarrow \ldots .$. majoring in Chemistry</p>
<p>$E_3 \rightarrow$. . . . . . . Mathematics</p>
<p>$\# \left(E_1\right)=20, \# \left(E_2\right)=20, \# \left(E_3\right)=10$</p>
<p>$\# \left(E_1 \times E_2 \times E_3\right)=20 \times 20 \times 10=4000$</p>
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<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-08-chapter-03-permutation-combination-L-3-5-lq5e9wj1jh8-12a.jpg"></p>
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### <Success>Example 2</Success>
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<p>A ternary sequence consists of digits $0,1,2$. How many 5-digit ternary sequences can be formed?</p>
<p><strong>Solution:</strong></p>
<p>$(.) \dot (.) \dot (.) \dot (.) \dot (.)$</p>
<p>In each position we can place either of 0,1 or 2</p>
<p>So the number of ways of filling each position is 3. Total no. of positions is 5 .</p>
<p>So by multiplication principle (MP), the total number of such ternary sequences is</p>
<p>$3 \times 3 \times 3 \times 3 \times 3=3^5=243$.</p>
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<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-08-chapter-03-permutation-combination-L-3-5-lq5e9wj1jh8-13a.jpg"></p>
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