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Solution


Q.

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______.           [2024]

Ans.

(70)

Number between 100 to 1000 are 3-digit number.

Let number abc such that a+b+c=14

where a,b,c∈{0,1,2,...9} and a≥1

Case I : All three digits are same ie., a=b=c

⇒3a=14, which is not possible

Case II : Two digits are same i.e., a=b≠c

⇒2a+c=14

∴  (a, c) = {(3, 8), (4, 6), (5, 4), (6, 2), (7, 0)}

In each subcase, number of ways of forming a 3-digit number = 3!2!

There are 5 such cases as (a, c) have 5 elements in which 0,7,7 is included

So, total number of 3 digits numbers when 2 digits are same 5×3!2!-1

=15-1=14


Case III : All digits are different

∴   (a, b, c) = {(1, 4, 9), (2, 4, 8), (2, 3, 9), (1, 5, 8), (3, 4, 7), (2, 5, 7), (1, 6, 7), (3, 5, 6), (5, 9, 0), (6, 8, 0)}


Total number of 3 digits number formed by above triplet = 10×3!-2×2!=60-4=56


Total number of required number = 56 + 14 = 70