Four marbles are drawn at random from a bag containing 7 black marbles and 10 red marbles. Determine the number of outcomes in the sample space of this experiment if the marbles are drawn without replacement.

All black --- 1

3black, 1red = 4!/(3!1!) = 4
2black, 2red = 4!(2!2!) = 6
1 black, 3red = 4
all red = 1

BBBB

BBBR
BBRB
BRBB
RBBB

BBRR
RRBB
BRBR
RBRB
BRRB
RBBR

RRRB
RRBR
RBRR
BRRR

RRRR

To determine the number of outcomes in the sample space, we need to calculate the number of ways we can choose 4 marbles out of the total 17 marbles without replacement.

The formula to calculate this is nCr, which stands for "n choose r" and is given by:

nCr = n! / (r!(n-r)!)

where n is the total number of items, and r is the number of items we want to choose.

In this case, n = 17 (the total number of marbles), and r = 4 (the number of marbles to be drawn).

Plugging the values into the formula:

17C4 = 17! / (4!(17-4)!)
= 17! / (4! * 13!)

Calculating the factorials:

17! = 17 * 16 * 15 * 14 * 13!

4! = 4 * 3 * 2 * 1

Substituting the factorials back into the equation:

17C4 = (17 * 16 * 15 * 14 * 13!) / (4 * 3 * 2 * 1 * 13!)

The 13! terms on both the numerator and denominator cancel out, leaving us with:

17C4 = (17 * 16 * 15 * 14) / (4 * 3 * 2 * 1)

Calculating the values:

17C4 = 2380

Therefore, there are 2380 possible outcomes in the sample space of this experiment when 4 marbles are drawn without replacement from a bag containing 7 black marbles and 10 red marbles.

To determine the number of outcomes in the sample space of this experiment when marbles are drawn without replacement, we need to consider the number of ways we can choose 4 marbles out of the total 17 marbles in the bag.

First, let's calculate the number of ways to choose 4 marbles out of the 17 marbles. We can use the binomial coefficient formula to find this.

The binomial coefficient formula, written as "n choose k," calculates the number of ways to choose k items from a set of n items. It is denoted by the symbol (n C k) or "${n \choose k}$". The formula for calculating the binomial coefficient is:

${n \choose k} = \frac{n!}{k!(n-k)!}$

Where "!" denotes a factorial, which means multiplying a number by all the positive integers less than it down to 1.

In this case, we want to select 4 marbles out of 17, so n = 17 and k = 4.

Let's calculate this:

${17 \choose 4} = \frac{17!}{4!(17-4)!}$

Simplifying:

${17 \choose 4} = \frac{17!}{4!13!} = \frac{17 \times 16 \times 15 \times 14}{4 \times 3 \times 2 \times 1}$

Calculating this:

${17 \choose 4} = 2380$

Therefore, the number of outcomes in the sample space of this experiment is 2380.