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A student has 10 blue pens, 5 red pens, and 7 black pens.
2) In how many ways can he select 5 pens if only the choice of colors matters.
Use the Counting Rules

A student has $10$ blue pens, $5$ red pens, and $7$ black pens. $\newline$ $2$ ) In how many ways can he select $5$ pens if only the choice of colors matters. $\newline$ Use the Counting Rules

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Solve a system of equations using elimination: word problems

Full solution

Q. A student has

10

blue pens,

5

red pens, and

7

black pens.

\newline

2

) In how many ways can he select

5

pens if only the choice of colors matters.

\newline

Use the Counting Rules

Identify Categories: Since only the choice of colors matters, we have three categories: blue, red, and black. We need to find the number of ways to choose $5$ pens from these categories.
List Combinations: We can have the following combinations of pens: $(3,1,1)$ , $(2,2,1)$ , $(2,1,2)$ , $(1,3,1)$ , $(1,2,2)$ , $(1,1,3)$ , $(0,2,3)$ , $(0,3,2)$ , $(2,0,3)$ , $(2,3,0)$ , $(2,2,1)$ $0$ , $(2,2,1)$ $1$ , $(2,2,1)$ $2$ , $(2,2,1)$ $3$ , $(2,2,1)$ $4$ , $(2,2,1)$ $5$ , $(2,2,1)$ $6$ , $(2,2,1)$ $7$ , $(2,2,1)$ $8$ , $(2,2,1)$ $9$ , $(2,1,2)$ $0$ . Each combination represents the number of blue, red, and black pens respectively.
Correct Mistake: However, we made a mistake in the previous step. We cannot have combinations like $(4,1,0)$ because we are only selecting $5$ pens, and the combinations must add up to $5$ . Let's correct this.

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