How many balls are in the bag? The bag contains only three types of colored balls: blue, red, and white.

Statement I: The ratio of blue balls to red balls is $4 : 5$. The number of white balls is one more than the number of red balls.

Statement II: The number of ways to select 2 red balls from the bag is 10.

Question

How many balls are in the bag? The bag contains only three types of colored balls: blue, red, and white.

Statement I: The ratio of blue balls to red balls is $4 : 5$. The number of white balls is one more than the number of red balls.

Statement II: The number of ways to select 2 red balls from the bag is 10.

Accepted Answer

**Solution:**

1. From Statement I:
   - Let the number of blue balls be $4x$ and red balls be $5x$.
   - Number of white balls = red balls + 1 = $5x + 1$.
   - Total balls = $4x + 5x + (5x + 1) = 14x + 1$.
   - However, the exact value of $x$ is unknown, so total balls cannot be determined.

2. From Statement II:
   - Number of ways to select 2 red balls = 10.
   - Using combination formula: $\binom{r}{2} = 10$, where $r$ is the number of red balls.
   - Solve for $r$:
     $$\frac{r(r-1)}{2} = 10 \implies r(r-1) = 20$$
     - Try $r=5$: $5 	imes 4 = 20$ which satisfies the equation.
   - So, red balls = 5.
   - But without information about blue and white balls, total balls cannot be found.

3. Combining both statements:
   - From Statement II, red balls = 5.
   - From Statement I, blue balls = $4x$, red balls = $5x$.
   - Since red balls = 5, $5x = 5 \implies x = 1$.
   - Blue balls = $4 	imes 1 = 4$.
   - White balls = red balls + 1 = $5 + 1 = 6$.
   - Total balls = $4 + 5 + 6 = 15$.

**Answer:** Both statements together are necessary to determine the total number of balls in the bag.

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