Power Play (Exponents) | A

Question 2

Think about how many combinations are possible in different contexts. Some examples are—

(i) Pincodes of places in India—The Pincode of Vidisha in Madhya Pradesh is 464001. The Pincode of Zemabawk in Mizoram is 796017.

(ii) Mobile numbers.

(iii) Vehicle registration numbers.

Try to find out how these numbers or codes are allotted/generated.

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Solution
Understand the Question
  • The total number of possible combinations in structured numbering systems is determined by the Fundamental Counting Principle (Multiplication Rule): multiply the number of available possibilities for each position.
  • For numerical digits (00 to 99), there are 1010 possible choices per position.
  • For English alphabets (A\text{A} to Z\text{Z}), there are 2626 possible choices per position.
  • Any specific restrictions (such as allowed starting digits) modify the number of choices for those specific positions.

(i) Pincodes of places in India—The Pincode of Vidisha in Madhya Pradesh is 464001. The Pincode of Zemabawk in Mizoram is 796017.

Step 1 · Calculate Theoretical PIN Code Combinations

A Postal Index Number (PIN) code consists of 66 digits:

  • 1st digit: Postal region
  • 2nd digit: Sub-region
  • 3rd digit: Sorting district
  • Last 3 digits: Specific post officeDiagram 1

Each of the 66 positions can be any digit from 00 to 99 (1010 choices each):

  • Number of choices for the 1st digit =10= 10
  • Number of choices for the 2nd digit =10= 10
  • Number of choices for the 3rd digit =10= 10
  • Number of choices for the 4th digit =10= 10
  • Number of choices for the 5th digit =10= 10
  • Number of choices for the 6th digit =10= 10
Total theoretical combinations=10×10×10×10×10×10=106\begin{aligned} \text{Total theoretical combinations} &= 10 \times 10 \times 10 \times 10 \times 10 \times 10 \\[0.6em] &= 10^6 \end{aligned}
Answer

(i) 1,000,000 combinations1{,}000{,}000\text{ combinations}

(ii) Mobile numbers.

Step 1 · Calculate Mobile Number Combinations

Indian mobile numbers contain 1010 digits and start with 6,7,8,6, 7, 8, or 99.Diagram 2

  • Number of choices for the 1st digit =4= 4
  • Number of choices for the 2nd digit =10= 10
  • Number of choices for the 3rd digit =10= 10
  • Number of choices for the 4th digit =10= 10
  • Number of choices for the 5th digit =10= 10
  • Number of choices for the 6th digit =10= 10
  • Number of choices for the 7th digit =10= 10
  • Number of choices for the 8th digit =10= 10
  • Number of choices for the 9th digit =10= 10
  • Number of choices for the 10th digit =10= 10
Total combinations=4×10×10×10×10×10×10×10×10×10=4×109\begin{aligned} \text{Total combinations} &= 4 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \\[0.6em] &= 4 \times 10^9 \end{aligned}
Answer

(ii) 4,000,000,000 numbers4{,}000{,}000{,}000\text{ numbers}

(iii) Vehicle registration numbers.

Step 1 · Calculate Vehicle Registration Combinations per RTO

A standard vehicle registration format (e.g., DL 01 AB 1234) has:

  • State/UT code (22 letters)
  • RTO code (22 digits)
  • Series (22 letters)
  • Unique vehicle number (44 digits)Diagram 3

For a specific RTO (fixed State and RTO code):

  • Number of choices for the 1st letter of the series =26= 26

  • Number of choices for the 2nd letter of the series =26= 26 Total series combinations=26×26=262\text{Total series combinations} = 26 \times 26 = 26^2

  • Number of choices for each of the 44 digits =10= 10 Total unique number combinations=10×10×10×10=104\text{Total unique number combinations} = 10 \times 10 \times 10 \times 10 = 10^4

Total combinations per RTO=(Letter combinations for series)×(Number combinations for unique number)=262×104=676×10,000\begin{aligned} \text{Total combinations per RTO} &= (\text{Letter combinations for series}) \times (\text{Number combinations for unique number}) \\[0.6em] &= 26^2 \times 10^4 \\[0.6em] &= 676 \times 10{,}000 \end{aligned}
Answer

(iii) 6,760,000 combinations6{,}760{,}000\text{ combinations} per RTO

Common Mistakes
  • Addition vs. Multiplication: Adding the possibilities (10+10+…10 + 10 + \dots) instead of multiplying them (10×10×…10 \times 10 \times \dots) across independent positions.
  • Ignoring Position Constraints: Assuming 1010 choices for all 1010 digits in mobile numbers instead of accounting for the 44 allowed starting digits (6,7,8,96, 7, 8, 9).
  • Confusing Letters and Digits: Using base-1010 for alphabetical series positions instead of 2626 letters (A–Z).

More questions in A

Q1

How many times can you fold it over and over?

Estu says “I heard that a sheet of paper can’t be folded more than 7 times”.

Roxie replies “What if we use a thinner paper, like a newspaper or a tissue paper?”

Try it with different types of paper and see what happens.

Q2

Think about how many combinations are possible in different contexts. Some examples are—

(i) Pincodes of places in India—The Pincode of Vidisha in Madhya Pradesh is 464001. The Pincode of Zemabawk in Mizoram is 796017.

(ii) Mobile numbers.

(iii) Vehicle registration numbers.

Try to find out how these numbers or codes are allotted/generated.

Q3

Tremendous in Ten!

Find a partner to play this game with. In 10 seconds, the person who writes a number or an expression, using only the digits 0-9 and arithmetic operations, that gives a number that is the larger between the two wins the round.

Below are some conditions that you may consider for different rounds.

(i) Exponents are not allowed. Only addition is allowed.

(ii) Exponents are not allowed. Only addition and multiplication are allowed.

(iii) Exponents are allowed. Only addition is allowed.

(iv) Exponents are allowed. Any arithmetic operation is allowed.

You can create your own conditions and/or involve more people to play together.

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