Seven white balls and three black balls are randomly placed in a row. What is the probability that no two black balls are placed adjacently?

Concept:

Probability:

• The probability of an event can be calculated by simply dividing the favorable number of outcomes by the total number of possible outcomes.
• The value of the probability of an event to happen can lie between 0 and 1.

Combination Formula:

 ⁿC_r = \(\rm \frac{n!}{r!(n-r)!}\)

nCr = number of combinations

n = total number of objects in the set

r =number of choosing objects from the set

Calculation:

The number of ways of arranging 10 balls in a row = ¹⁰ P₁₀.​=10!

Now, the condition is to not arrange two black balls adjacently.

So, first, arrange the 7 while balls first and arrange these 3 black balls in between the created 8 places between each white ball.

So, number of favorable outcomes of arranging 3 black balls and 7 white balls

= 8 P3 × 7!

Thus, the probability that no two black balls are placed adjacently is
P = (8 P3 × 7!)​ / 10 P10

⇒ P = \(\rm \frac{7}{15}\)

∴ The required probability is \(\rm \frac{7}{15}\).