Question

There are an unlimited number of identical balls of three different colors. How many arrangements of almost 7 balls in a row can be made by using them?
A) 2180
B) 343
C) 399
D) 3279

Answer 1

Hint:
In the above question, for every blank you have 3 options A, B, C, there are 7 blanks, number of possible arrangements =${3^7}$ = 2187 which will be equal to if you take each case individually and calculate the number of cases.

Complete step by step solution:
Given, in the question we have an unlimited number of identical balls of three different colors.
Now,
There is an unlimited number of identical three rows.
Number of arrangements of 7 balls in a row = ${3^7}$
Number of arrangements of 6 balls in a row = ${3^6}$
Number of arrangements of 5 balls in a row = ${3^5}$
Number of arrangements of 4 balls in a row = ${3^4}$
Number of arrangements of 3 balls in a row = ${3^3}$
Number of arrangements of 2 balls in a row = ${3^2}$
Number of arrangements of 1 ball in a row = 3
Therefore, number of arrangements of almost 7 balls in row that can be made (n)=$3 + {3^2} + {3^3} + {3^4} + {3^5} + {3^6} + {3^7}$\[\]
As we can see it is in GP. We will use GP because we can see here that there is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed number.
$
  n = \dfrac{{3({3^7} - 1)}}{{3 - 1}} \\
   \Rightarrow \dfrac{3}{2}(2187 - 1) \\
   \Rightarrow \dfrac{3}{2} \times 2186 \\
   \Rightarrow 3 \times 1093 \\
   \Rightarrow n = 3279 \\
$

Option D is the correct answer.

Note:
We know that we have three different colors of balls. We have 7 places for each color. You should be careful while calculating because it seems too lengthy. So, that is why we will use the concept of Geometric Progression.