Francisco bought $12$ plants to arrange along the border of his garden. How many distinct arrangements can he make if the plants are comprised of $6$ tulips, $3$ roses, and $3$ daisies?

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Grade 8

Experimental probability

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Q. Francisco bought

12

plants to arrange along the border of his garden. How many distinct arrangements can he make if the plants are comprised of

6

tulips,

3

roses, and

3

daisies?

Calculate Factorials: We need to calculate the number of distinct arrangements Francisco can make with his $12$ plants, which include $6$ tulips, $3$ roses, and $3$ daisies. This is a permutation problem involving identical items. The formula for the number of distinct arrangements (permutations) of $n$ items where there are $n_1$ identical items of one type, $n_2$ identical items of another type, and so on, is given by: $\newline$ $P = \frac{n!}{(n_1! \cdot n_2! \cdot \ldots \cdot n_k!)}$ $\newline$ where $n!$ represents the factorial of $n$ , and $n_1!$ , $6$ $0$ , ..., $6$ $1$ represent the factorials of the number of identical items.
Calculate Factorials for Each Group: First, let's calculate the factorial of the total number of plants, which is $12!$ ( $12$ factorial). $\newline$ $12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Apply Permutation Formula: Next, we calculate the factorial for each group of identical plants: $\newline$ For the $6$ tulips, we have $6!$ ( $6$ factorial). $\newline$ $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $\newline$ For the $3$ roses, we have $3!$ ( $3$ factorial). $\newline$ $3! = 3 \times 2 \times 1$ $\newline$ For the $3$ daisies, we have $3!$ ( $3$ factorial). $\newline$ $3! = 3 \times 2 \times 1$
Simplify Expression: Now, we apply the formula to find the number of distinct arrangements: $\newline$ $P = \frac{12!}{6! \times 3! \times 3!}$ $\newline$ We already calculated the factorials, so we can substitute them into the formula: $\newline$ $P = \frac{(12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{((6 \times 5 \times 4 \times 3 \times 2 \times 1) \times (3 \times 2 \times 1) \times (3 \times 2 \times 1))}$
Perform Division: We can simplify the expression by canceling out common factors in the numerator and the denominator: $\newline$ P = $(12 \times 11 \times 10 \times 9 \times 8 \times 7) / ((3 \times 2 \times 1) \times (3 \times 2 \times 1))$ $\newline$ P = $(12 \times 11 \times 10 \times 9 \times 8 \times 7) / (6 \times 6)$ $\newline$ P = $(12 \times 11 \times 10 \times 9 \times 8 \times 7) / 36$
Perform Division: We can simplify the expression by canceling out common factors in the numerator and the denominator: $\newline$ $P = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{(3 \times 2 \times 1) \times (3 \times 2 \times 1)}$ $\newline$ $P = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 6}$ $\newline$ $P = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{36}$ Now, we perform the division to get the final answer: $\newline$ $P = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{36}$ $\newline$ $P = \frac{2 \times 11 \times 10 \times 9 \times 8 \times 7}{1}$ $\newline$ $P = 11 \times 10 \times 9 \times 8 \times 7$ $\newline$ $P = 55440$