Permutations and Combinations 2 Question 3
3. Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is
(2019 Main, 9 April II)
(a) 262
(b) 190
(c) 225
(d) 157
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Answer:
Correct Answer: 3. (b)
Solution:
- Let there are $n$ balls used to form the sides of equilateral triangle.
According to the question, we have
$$ \begin{aligned} & \frac{n(n+1)}{2}+99=(n-2)^{2} \\ \Rightarrow & n^{2}+n+198=2\left[n^{2}-4 n+4\right] \\ \Rightarrow & n^{2}-9 n-190=0 \\ \Rightarrow & n^{2}-19 n+10 n-190=0 \\ \Rightarrow & (n-19)(n+10)=0 \\ \Rightarrow & n=19,-10 \\ \Rightarrow & n=19 \quad[\because \text { number of balls } n>0] \end{aligned} $$
Now, number of balls used to form an equilateral triangle is $\frac{n(n+1)}{2}$
$$ =\frac{19 \times 20}{2}=190 $$
