ct as mentor and explain . How many 6 digit numbers can be formed from digits 2,2,3,3,4,4,? How many of them will lie between 400000 and 430000 ?

Sure, let's break this down:

1. First, we need to determine how many 6-digit numbers can be formed from the digits 2,2,3,3,4,4. This is a permutation problem with repetition. The formula for permutations with repetition is n!/r1!r2!...rn!, where n is the total number of items, and r1, r2, ..., rn are the numbers of times each item is repeated. In this case, n=6 (because we have 6 digits), r1=2 (because the digit 2 is repeated twice), r2=2 (because the digit 3 is repeated twice), and r3=2 (because the digit 4 is repeated twice). So the total number of 6-digit numbers is 6!/(2!2!2!) = 90.

2. Next, we need to determine how many of these numbers lie between 400000 and 430000. For a number to be in this range, it must start with the digit 4. After that, we have 5 digits left to arrange, which can be any of the remaining 5 digits (2,2,3,3,4). This is again a permutation problem with repetition. In this case, n=5 (because we have 5 digits left), r1=2 (because the digit 2 is repeated twice), r2=2 (because the digit 3 is repeated twice), and r3=1 (because the digit 4 is repeated once). So the total number of numbers in this range is 5!/(2!2!1!) = 30.

So, there are 90 total 6-digit numbers that can be formed from the digits 2,2,3,3,4,4, and 30 of these lie between 400000 and 430000.