Number of all possible ways of distributing eight identical apples among three persons is
Show Hint
Remember the "stars and bars" method for distribution problems. For $n$ identical items and $k$ distinct recipients, imagine the $n$ items as 'stars' (*) and you need $k-1$ 'bars' (|) to divide them into $k$ groups. The total number of arrangements of these $n$ stars and $k-1$ bars is the answer, which is $^{n+k-1}C_{k-1}$.
Step 1: Identify Method
This is a problem of distributing \( n \) identical items into \( r \) distinct groups (persons).
Formula (Stars and Bars): \( ^{n+r-1}C_{r-1} \).
Step 2: Calculate
Given \( n = 8 \) (apples) and \( r = 3 \) (persons).
Ways = \( ^{8+3-1}C_{3-1} = ^{10}C_2 \).
\[ ^{10}C_2 = \frac{10 \times 9}{2} = 45 \]