<p class="paragraph-header">Addition principle — count by different cases</p>
<p>Suppose a task can be completed by mutually exclusive plans \(A_1,A_2,\ldots,A_n\), with \(a_1,a_2,\ldots,a_n\) ways for the respective plans. Since exactly one plan is used, add the numbers of ways.</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-left" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><div class="content-box-title">Addition principle</div><p>\[\sum\limits_{i=1}^{n}a_i=a_1+a_2+\cdots+a_n\]</p></div></div>
<p class="paragraph-header">Multiplication principle — count by different steps</p>
<p>Suppose a task is completed through successive steps \(B_1,B_2,\ldots,B_n\), with \(b_i\) available choices at step \(i\) for every allowed sequence of earlier choices. Multiply these numbers of choices. The choices need not be independent; the branch count at each step must be the same for every earlier choice.</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-left" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><div class="content-box-title">Multiplication principle</div><p>\[b_1b_2\cdots b_n\]</p></div></div>

Addition principle — count by different cases

Suppose a task can be completed by mutually exclusive plans \(A_1,A_2,\ldots,A_n\), with \(a_1,a_2,\ldots,a_n\) ways for the respective plans. Since exactly one plan is used, add the numbers of ways.

\[\sum\limits_{i=1}^{n}a_i=a_1+a_2+\cdots+a_n\]

Addition principle

Multiplication principle — count by different steps

Suppose a task is completed through successive steps \(B_1,B_2,\ldots,B_n\), with \(b_i\) available choices at step \(i\) for every allowed sequence of earlier choices. Multiply these numbers of choices. The choices need not be independent; the branch count at each step must be the same for every earlier choice.

Fundamental Counting Principles

Fundamental Counting Principles