Full of two's and five's

Probability Level 3

How many numbers of three digits are such that have at least one 2 and one 5?

The answer is 52.

1 solution

Paola Ramírez
Apr 18, 2015

Case 1: A number of formed by ${b,b,a}$ only can be arrangement of three ways so we have two posibilities $b=5, 2$ . $\boxed{6}$

Case 2: Three different digits ${a,b,c}$

Let $a=2$ and $b=5$ , this numbers can be arrangement in $\binom{3}{2}$ ways and permuted of two forms, $a$ and $b$ can be arrangement of $6$ forms.

Then $c=8$ because you cannot repeat $5$ and $2$ . Also delete arrangements $052$ and $025$

By this case we have $6\times 8-2=\boxed{46}$

Total of numbers $6+46=\boxed{52}$

I keep getting 452 as my answer.

There are 9x10x10 = 900 three-digit numbers and there are 7x8x8 = 448 numbers that don't have any digits that are either a 2 or 5. Thus, there should be 900 - 448 = 452 three-digit numbers that have at least one digit that is either a 2 or 5.

This can also be confirmed with a Python program:

count = 0
for a in range(100, 1000):
  if ("2" in list(str(a))) or ("5" in list(str(a))):
                               count+=1

print (count)

Again, the program returns 452 as the answer.

Mark Mottian - 6 years, 1 month ago

Full of two's and five's

The answer is 52.

1 solution

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