How many two-digit integers do not have any digits that are equal to

Question

How many positive two-digit integers do not have any digits that are equal to each other?

(A) 90
(B) 81
(C) 80
(D) 75
(E) 72

This is a  PS Butler Question

Check the links to other Butler Projects:

Data Sufficiency Butler  
  Critical Reasoning Butler  
  Reading Comprehension Butler  
  Data Insights Butler

hr1212 · Answer

Total two digit numbers = 90
Two digit numbers with equal digits = 9 (11,22,33,...,99)

Numbers with no equal digits = 90 - 9 = 81

IMO: B

Whoisdmx15 · Answer

Two digit numbers , with no digit repeating:

_ * _

Tenths digit can be filled by 9 digits (all except 0)
Units digit can be filled by 9 digits (all including 0 except the first digit selected)
Hence total two digit numbers: 9 * 9 = 81
Hence  IMO Ans B

eega · Answer

Number of two digit intergers = 90 (10 - 99)
Number of two digit integers with same digit = 9 (11 - 99)

Ans = 90-9 = 81  Option B

areyPras · Answer

In this question, it just says integers, so shouldn't we consider negative integers as well?

Bunuel · Answer

Yes,  positive  should have been mentioned there. Edited. Thank you.

ThenSweat · Answer

We are counting **two-digit numbers** where the two digits are **different**.

Examples that work:

* **12** ✅
* **47** ✅
* **90** ✅

Examples that do **not** work:

* **11** ❌ (both digits are the same)
* **55** ❌
* **99** ❌

Think of a two-digit number as having:

```text id="j4p4rq"
   
```

### Step 1: Choose the first digit

The first digit cannot be **0**, because then it would not be a two-digit number.

So we can choose:

```text id="2m9o5e"
1, 2, 3, 4, 5, 6, 7, 8, or 9
```

That gives us **9 choices**.

### Step 2: Choose the second digit

There are **10 possible digits**:

```text id="v6gqcy"
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
```

But the second digit **cannot equal the first digit**.

So:

```text id="hmf2jp"
10 − 1 = 9 choices
```

### Step 3: Multiply

For every one of the **9 choices** for the first digit, there are **9 choices** for the second digit:

```text id="hbrdf4"
9 × 9 = 81
```

✅ **Answer: (B) 81**

Another quick way:

There are **90 two-digit numbers** total:

```text id="i7hyy7"
10 through 99 = 90 numbers
```

There are **9 numbers with matching digits**:

```text id="v0sw79"
11, 22, 33, 44, 55, 66, 77, 88, 99
```

So:

```text id="ddf3ta"
90 − 9 = 81
```

Therefore, **81 two-digit integers have different digits**.

Rad29 · Answer

Tens digit can be filled in 9 ways (Tens digit can't be 0) and units place can be filled in 9 ways (excluding the digit chosen for tens but including 0) - 9*9=81

How many two-digit integers do not have any digits that are equal to

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

How many two-digit integers do not have any digits that are equal to

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards