Question $$ \mathbf{2 0} $$ In the numerical representation of dates, January 23,2024 , is expressed as 20240123, and September 2, 2024, is denoted as 20240902 . How many occurrences of the digit ' 2 ' exist in all 8 -digit numbers representing dates in the year 2024 ?

Question

UpStudy AI · Accepted Answer

We write each date in the year 2024 in the 8‐digit form  
$$
\texttt{YYYYMMDD},
$$  
and then we count the occurrences of the digit $$2$$ in all valid dates.

**Step 1. The Year Part $$2024$$**

Every date in 2024 begins with the fixed 4 digits “2024”.  
In “2024” the digit $$2$$ appears twice (in the first and third positions).  
Since there are 366 dates in 2024 (a leap year), the total contribution from the year part is  
$$
2\times366=732.
$$

**Step 2. The Month Part**

Months are written as two digits from “01” to “12”. List the months with their digits and count the $$2$$’s:

- Month “01”: digits “0” and “1” $$\rightarrow$$ 0 occurrences.
- Month “02”: digits “0” and “2” $$\rightarrow$$ 1 occurrence.
- Month “03”: “0” and “3” $$\rightarrow$$ 0.
- Month “04”: “0” and “4” $$\rightarrow$$ 0.
- Month “05”: “0” and “5” $$\rightarrow$$ 0.
- Month “06”: “0” and “6” $$\rightarrow$$ 0.
- Month “07”: “0” and “7” $$\rightarrow$$ 0.
- Month “08”: “0” and “8” $$\rightarrow$$ 0.
- Month “09”: “0” and “9” $$\rightarrow$$ 0.
- Month “10”: “1” and “0” $$\rightarrow$$ 0.
- Month “11”: “1” and “1” $$\rightarrow$$ 0.
- Month “12”: “1” and “2” $$\rightarrow$$ 1 occurrence.

Only month “02” and month “12” contribute.\  
Now, count how many valid days are in these months:

- For February (“02”), 2024 has 29 days.
- For December (“12”), there are 31 days.

Thus, the total contributions from the month parts are  
$$
\text{February: } 29\times 1 = 29\[1mm]
\text{December: } 31\times 1 = 31.
$$
So, total from month parts is  
$$
29+31=60.
$$

**Step 3. The Day Part**

For each date, the day (DD) is represented in two digits with a leading zero if needed. The number of days depends on the month. We must sum the number of $$2$$’s in the two-digit representation for each valid day.

Let’s determine the count $$f(n)$$ for day numbers from $$01$$ to $$n$$ (where $$n\ge29$$):

- For days $$01$$ to $$09$$:  
  Only “02” contains a $$2$$, contributing $$1$$.

- For days $$10$$ to $$19$$:  
  Only “12” contains a $$2$$, contributing $$1$$.

- For days $$20$$ to $$29$$:  
  Every day has a $$2$$ in the tens place (contributing $$1$$ each for 10 days) and in the special case “22” the ones digit is also $$2$$.  
  So the total contribution here is  
  $$
  10 + 1 = 11.
  $$

- For days $$30$$ (and $$31$$ when applicable):  
  “30” and “31” do not include the digit $$2$$.

Thus, whether a month has 29, 30, or 31 days, the sum for the day part is  
$$
f(n)=1+1+11=13.
$$

Since every month in 2024 has at least 29 days (February has 29, and the others have 30 or 31 days), the day part from each month contributes $$13$$ occurrences of the digit $$2$$.

There are 12 months, so the total from the day parts is  
$$
12\times 13 =156.
$$

**Step 4. Total Occurrences**

Now sum the contributions from all parts:

- Year part: $$732$$
- Month part: $$60$$
- Day part: $$156$$

Thus, the total number of occurrences of the digit $$2$$ is  
$$
732+60+156=948.
$$

The final answer is: $$948$$.
There are 948 occurrences of the digit '2' in all 8-digit date numbers representing dates in the year 2024.

Question
In the numerical representation of dates, January 23,2024 , is expressed as
20240123, and September 2, 2024, is denoted as 20240902 . How many occurrences
of the digit ’ 2 ’ exist in all 8 -digit numbers representing dates in the year 2024 ?

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Arithmetic Questions

Question In the numerical representation of dates, January 23,2024 , is expressed as 20240123, and September 2, 2024, is denoted as 20240902 . How many occurrences of the digit ’ 2 ’ exist in all 8 -digit numbers representing dates in the year 2024 ?