{"id":1750,"date":"2025-07-28T19:01:52","date_gmt":"2025-07-28T19:01:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1750"},"modified":"2026-03-26T16:55:38","modified_gmt":"2026-03-26T16:55:38","slug":"counting-principles-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/counting-principles-learn-it-3\/","title":{"raw":"Counting Principles: Learn It 3","rendered":"Counting Principles: Learn It 3"},"content":{"raw":"

Finding the Number of Permutations of [latex]n[\/latex] Non-Distinct Objects<\/h2>\r\nWe have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable?\r\n\r\n
For example, suppose there is a sheet of [latex]12[\/latex] stickers. If all of the stickers were distinct, there would be [latex]12![\/latex] ways to order the stickers.\r\n[latex]\\\\[\/latex]\r\nHowever, [latex]4[\/latex] of the stickers are identical stars, and [latex]3[\/latex] are identical moons. Because all of the objects are not distinct, many of the [latex]12![\/latex] permutations we counted are duplicates.\r\n[latex]\\\\[\/latex]\r\nThis means we need to divide by the number of ways to order the [latex]4[\/latex] stars and the ways to order the [latex]3[\/latex] moons to find the number of unique permutations of the stickers. There are [latex]4![\/latex] ways to order the stars and [latex]3![\/latex] ways to order the moon.\r\n

[latex]\\dfrac{12!}{4!3!}=3\\text{,}326\\text{,}400[\/latex]<\/p>\r\nThere are [latex]3,326,400[\/latex] ways to order the sheet of stickers.\r\n\r\n<\/section>

\r\n

formula for finding the number of permutations of [latex]n[\/latex] non-distinct objects<\/h3>\r\nIf there are [latex]n[\/latex] elements in a set and [latex]{r}_{1}[\/latex] are alike, [latex]{r}_{2}[\/latex] are alike, [latex]{r}_{3}[\/latex] are alike, and so on through [latex]{r}_{k}[\/latex], the number of permutations can be found by\r\n

[latex]\\dfrac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}[\/latex]<\/p>\r\n\r\n<\/section>

Find the number of rearrangements of the letters in the word DISTINCT.[reveal-answer q=\"162138\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162138\"]\r\n
    \r\n \t
  • There are [latex]8[\/latex] letters.<\/li>\r\n \t
  • Both I and T are repeated [latex]2[\/latex] times.<\/li>\r\n<\/ul>\r\nSubstitute [latex]n=8, {r}_{1}=2, [\/latex] and [latex] {r}_{2}=2 [\/latex] into the formula:\r\n

    [latex]\\dfrac{8!}{2!2!}=10\\text{,}080 [\/latex]<\/p>\r\nThere are [latex]10,080[\/latex] arrangements.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321992[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]321993[\/ohm_question]<\/section>","rendered":"

    Finding the Number of Permutations of [latex]n[\/latex] Non-Distinct Objects<\/h2>\n

    We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable?<\/p>\n

    For example, suppose there is a sheet of [latex]12[\/latex] stickers. If all of the stickers were distinct, there would be [latex]12![\/latex] ways to order the stickers.
    \n[latex]\\\\[\/latex]
    \nHowever, [latex]4[\/latex] of the stickers are identical stars, and [latex]3[\/latex] are identical moons. Because all of the objects are not distinct, many of the [latex]12![\/latex] permutations we counted are duplicates.
    \n[latex]\\\\[\/latex]
    \nThis means we need to divide by the number of ways to order the [latex]4[\/latex] stars and the ways to order the [latex]3[\/latex] moons to find the number of unique permutations of the stickers. There are [latex]4![\/latex] ways to order the stars and [latex]3![\/latex] ways to order the moon.<\/p>\n

    [latex]\\dfrac{12!}{4!3!}=3\\text{,}326\\text{,}400[\/latex]<\/p>\n

    There are [latex]3,326,400[\/latex] ways to order the sheet of stickers.<\/p>\n<\/section>\n

    \n

    formula for finding the number of permutations of [latex]n[\/latex] non-distinct objects<\/h3>\n

    If there are [latex]n[\/latex] elements in a set and [latex]{r}_{1}[\/latex] are alike, [latex]{r}_{2}[\/latex] are alike, [latex]{r}_{3}[\/latex] are alike, and so on through [latex]{r}_{k}[\/latex], the number of permutations can be found by<\/p>\n

    [latex]\\dfrac{n!}{{r}_{1}!{r}_{2}!\\dots {r}_{k}!}[\/latex]<\/p>\n<\/section>\n

    Find the number of rearrangements of the letters in the word DISTINCT.<\/p>\n