Question Details

How many pair of digits can be formed in the number 73915243, which have same number of digits between then as in the number series (both forward and backward direction)?

Options

A

None

B

Two

C

Three

D

More than three

Show Answer

Correct Answer :

Option D

More than three

More than three

Solution :

The correct option is More than three.

To find the pairs of digits in the number 73915243 that have the same number of digits between them as in the standard number series (in both forward and backward directions), we analyze the positions of each digit.
Let us write down the number and index its digits from left to right:
Digit: 7 (index 1), 3 (index 2), 9 (index 3), 1 (index 4), 5 (index 5), 2 (index 6), 4 (index 7), 3 (index 8).

Let us examine each pair in both the forward and backward directions. A pair of digits (A, B) at positions i and j (where i<j) has the same number of digits between them as in the numerical sequence if the absolute difference between the digits is equal to the difference between their indices:
|A-B|=j-i

Let us perform the checks:
1. Forward Direction:
- From 7 (index 1):
- 7 to 3 (index 2): difference is 1, digit difference is 4. No. - 7 to 9 (index 3): difference is 2, digit difference is 2. No. - 7 to 1 (index 4): difference is 3, digit difference is 6. No. - 7 to 5 (index 5): difference is 4, digit difference is 2. No. - 7 to 2 (index 6): difference is 5, digit difference is 5. Yes! The pair is (7, 2). There are 4 digits (3, 9, 1, 5) between them in the number, and in the number series, there are also 4 digits (6, 5, 4, 3) between 7 and 2. - 7 to 4 (index 7): difference is 6, digit difference is 3. No. - 7 to 3 (index 8): difference is 7, digit difference is 4. No.
- From 3 (index 2): - 3 to 9 (index 3): difference is 1, digit difference is 6. No. - 3 to 1 (index 4): difference is 2, digit difference is 2. Yes! The pair is (3, 1). There is 1 digit (9) between them, and in the number series, 2 is between 3 and 1. - 3 to 5 (index 5): difference is 3, digit difference is 2. No. - 3 to 2 (index 6): difference is 4, digit difference is 1. No. - 3 to 4 (index 7): difference is 5, digit difference is 1. No. - 3 to 3 (index 8): difference is 6, digit difference is 0. No.
- From 9 (index 3): - 9 to 1 (index 4): difference is 1, digit difference is 8. No. - 9 to 5 (index 5): difference is 2, digit difference is 4. No. - 9 to 2 (index 6): difference is 3, digit difference is 7. No. - 9 to 4 (index 7): difference is 4, digit difference is 5. No. - 9 to 3 (index 8): difference is 5, digit difference is 6. No.
- From 1 (index 4): - 1 to 5 (index 5): difference is 1, digit difference is 4. No. - 1 to 2 (index 6): difference is 2, digit difference is 1. No. - 1 to 4 (index 7): difference is 3, digit difference is 3. Yes! The pair is (1, 4). There are 2 digits (5, 2) between them in the number, and in the number series, 2 and 3 are between 1 and 4. - 1 to 3 (index 8): difference is 4, digit difference is 2. No.
- From 5 (index 5): - 5 to 2 (index 6): difference is 1, digit difference is 3. No. - 5 to 4 (index 7): difference is 2, digit difference is 1. No. - 5 to 3 (index 8): difference is 3, digit difference is 2. No.
- From 2 (index 6): - 2 to 4 (index 7): difference is 1, digit difference is 2. No. - 2 to 3 (index 8): difference is 2, digit difference is 1. No.
- From 4 (index 7): - 4 to 3 (index 8): difference is 1, digit difference is 1. Yes! The pair is (4, 3). There are 0 digits between them in both.

2. Backward Direction:
Let us check backward pairs:
- From 3 (index 8): - 3 to 4 (index 7): difference is 1, digit difference is 1. Already counted as (4, 3) / (3, 4). - 3 to 2 (index 6): difference is 2, digit difference is 1. No. - 3 to 5 (index 5): difference is 3, digit difference is 2. No. - 3 to 1 (index 4): difference is 4, digit difference is 2. No. - 3 to 9 (index 3): difference is 5, digit difference is 6. No. - 3 to 3 (index 2): difference is 6, digit difference is 0. No. - 3 to 7 (index 1): difference is 7, digit difference is 4. No.
- From 4 (index 7): - 4 to 2 (index 6): difference is 1, digit difference is 2. No. - 4 to 5 (index 5): difference is 2, digit difference is 1. No. - 4 to 1 (index 4): difference is 3, digit difference is 3. Already counted as (1, 4). - 4 to 9 (index 3): difference is 4, digit difference is 5. No. - 4 to 3 (index 2): difference is 5, digit difference is 1. No. - 4 to 7 (index 1): difference is 6, digit difference is 3. No.
- From 2 (index 6): - 2 to 5 (index 5): difference is 1, digit difference is 3. No. - 2 to 1 (index 4): difference is 2, digit difference is 1. No. - 2 to 9 (index 3): difference is 3, digit difference is 7. No. - 2 to 3 (index 2): difference is 4, digit difference is 1. No. - 2 to 7 (index 1): difference is 5, digit difference is 5. Already counted as (7, 2).
- From 5 (index 5): - 5 to 1 (index 4): difference is 1, digit difference is 4. No. - 5 to 9 (index 3): difference is 2, digit difference is 4. No. - 5 to 3 (index 2): difference is 3, digit difference is 2. No. - 5 to 7 (index 1): difference is 4, digit difference is 2. No.
- From 1 (index 4): - 1 to 9 (index 3): difference is 1, digit difference is 8. No. - 1 to 3 (index 2): difference is 2, digit difference is 2. Already counted as (3, 1). - 1 to 7 (index 1): difference is 3, digit difference is 6. No.
- From 9 (index 3): - 9 to 3 (index 2): difference is 1, digit difference is 6. No. - 9 to 7 (index 1): difference is 2, digit difference is 2. Yes! The pair is (9, 7). There is 1 digit (3) between them, and in the number series, 8 is between 9 and 7.

The valid pairs found are:
1. (7, 2) (Forward)
2. (3, 1) (Forward)
3. (1, 4) (Forward)
4. (4, 3) (Forward)
5. (9, 7) (Backward)

Since we have found 5 such pairs, which is more than 3, the correct option is More than three.

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