GO Classes Test Series 2024 | Discrete Mathematics | Test 4 | Question: 11

Answer 1

There are $1 \cdot 9 \cdot 8 \cdot 7 + 4 \cdot 8 \cdot 8 \cdot 7 = 2, 296.$ The trick is to note that there are two different ways to make these numbers: count the numbers that end in zero separately from the numbers that don’t end in zero.

• Let $\text{X}$ be the set of four-digit numbers with distinct digits that end in zero. There are $1 \cdot 9 \cdot 8 \cdot 7$ of them. The recipe is:
  1. Choose the last digit (which must be zero) – $1$ choice
  2. Choose the first digit (can’t be zero) – $9$ choices
  3. Choose the second digit (can’t be zero or the same as the first digit)– $8$ choices
  4. Choose the third digit (can’t be the same as either the first, second, or last) – $7$ choices

So $|\text{X}| = 1 \cdot 9 \cdot 8 \cdot 7.$

• Let $\text{Y}$ be the set of four-digit numbers with distinct digits and that don’t end in zero. There are $4 \cdot 8 \cdot 8 \cdot 7$ of them. The recipe is:
  1. Choose the last digit (which must be even and can’t be zero) – $4$ choices
  2. Choose the first digit (can’t be zero and can’t be the same as the last digit) – $8$ choices
  3. Choose the second digit (can’t be the same as the first or last digit)– $8$ choices
  4. Choose the third digit (can’t be the same as either the first, second, or last) – $7$ choices

So $|\text{Y}| = 4 \cdot 8 \cdot 8 \cdot 7.$

The set we are interested in is $\text{X} \cup \text{Y}.$ Since $\text{X} \cap  \text{Y} = \varnothing ,$

we see that $|\text{X} \cup \text{Y} | = |\text{X}| + |\text{Y}|.$

$\textbf{Alternate solution:}$

Let $\text{S}$ be the set of even four-digit integers with distinct digits.

Let $\text{T}$ be the set of odd four-digit integers with distinct digits.

We see that $|\text{T}| = 5 \cdot 8 \cdot 8 \cdot 7.$

We see that $|\text{T} \cup \text{S}| = 9 · 9 · 8 · 7.$

Note that $\text{T} \cap \text{S} = \varnothing .$

Thus $|\text{S}| = |\text{T} \cup \text{S}| - |\text{T}|.$

Hence $|\text{S}| = 9 \cdot 9 \cdot 8 \cdot 7 - 5 \cdot 8 \cdot 8 \cdot 7 = 2, 296.$

Comments

@Deepak Poonia Sir check my(genius) argument soln. is it correct ?

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@Deepak Poonia Sir for |TUS| the number of elements will be  _8_* _8_*_7_*_10_ i.e 4480 solutions because for the last place we will have 10 choices i.e 0 to 9. ??

And why are we separatly calculating for last digit 0 and last digit not 0 in solution.

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@abir_banerjee See what is $S \cup T.$

$S \cup T$ is Set of All 4-digits integers with distinct digits. So, since a 4-digit integer cannot start with 0, So, for left most position we have $9$ choice, then for next position we have $9$ choices because we can use 0 now for second position from left but we cannot use the digit we used for left most position.

So, $|T \cup S| = 9.9.8.7$

@Genius, Is the number $7206$ an even number with distinct 4-digits ?

If Yes, you have your answer.

The given answer is correct.

Answer 2

if last digit is 0,

9*8*7*4*1

if last digit is any of 2,4,6,8,

8*8*7*4

so, total = 9*8*7*1 + 8*8*7*4

              = 2296

Answer 3

@GO Classes soln is wrong for this question bcz its is asking for positive integers but 0 is neither positive nor negative.

so in that case answer will be 8*7*6*4=1334

Comments

at which place  are you placing the 0??

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@Genius, Is the number $7206$ an even number with distinct 4-digits ?

If Yes, you have your answer.

The given answer by @GO Classes is correct.

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sir why u taking 0 in consideration for digit as it is given positive integers so the digits must be 1 to 9 not 0 to 9?

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@Genius

Is 102 a Positive Integer?