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Question:

When opening his fruit shop for the day a shopkeeper found that his stock of apples could be perfectly arranged in a complete triangular array: that is, every row with one apple more than the row immediately above, going all the way up ending with a single apple at the top.

During any sales transaction, apples are always picked from the uppermost row, and going below only when that row is exhausted. When one customer walked in the middle of the day she found an incomplete array in display having 126 apples totally. How many rows of apples (complete and incomplete) were seen by this customer? (Assume that the initial stock did not exceed 150 apples.)

Explanation:

In the given triangular array, 1st row will have 1 apple, 2nd row will have 2 apples and so on.

∴ n^th row will have n apples.

⇒ Total number of apples initially = 1 + 2 + 3 + … + n = $\frac{n(n+1)}{2}$

Initially total number of apples should be greater than 126 but less than or equal to 150

∴ 126 < $\frac{n(n+1)}{2}$ ≤ 150

Only integral value of n satisfying this is 16.

⇒ $\frac{16\times 17}{2}$ = 136

∴ Initially there are 16 rows and 136 apples arranged in a triangular array in these 16 rows.

As the customer observes 126 apples, it would mean that 10 apples were removed. 

Now, apples are removed starting from Row 1, which has 1 apple. So apples must have been removed in such a way with 1 apple from Row 1, 2 apples from Row 2, 3 apples from Row 3 and 4 apples from Row 4.

∴ Total 10 apples were removed the first 4 rows.

As out of 16 rows, apples are removed till Row 4, it would mean that only 16 - 4 or 12 rows of apples are visible to the customer.

Hence, option (e).