<image>A box contains a large number of small balls that are all the same shape and size but have different weights. From this box, $$50$$ balls are randomly selected as samples, and their weights (in grams) are measured. The weight intervals are $$[5,15]$$, $$(15,25]$$, $$(25,35]$$, and $$(35,45]$$. Based on these measurements, a sample weight frequency distribution histogram is generated ($$as shown in the figure$$).
$$(1)$$ Find the value of $$a$$, and based on the sample data, estimate the mode and mean weight of the small balls in the box;
$$(2)$$ Randomly select $$3$$ balls from the box. Let $$X$$ be the number of balls whose weights are in the interval $$[5,15]$$. Find the distribution and the expectation of $$X$$. Both should be determined based on the frequencies in the histogram interpreted as probabilities.