Question

A four digit number plate of a car is said to be lucky if the sum of the first two digits is equal to the sum of the last two digit. Then find the total number of lucky plates. (Assume 0000, 0011, 0111 … all are four digit numbers)

Answer 1

Hint: In a number plate each digit of the plate ranges from 0 to 9 and since in this question first two digits is equal to sum of last two digit then we will check all the possibilities of the numbers by substituting the values one by one.

Complete step-by-step answer:
Let the 4 digits on a number plate be P, Q, R, S
Now it is said that sum of first two digits is equal to sum of last two digit and on a number plate the maximum sum of the two digits can be 18, so the number of possible number plate
When
\[P + Q = R + S = 0\]
Where LM, NO= 00
So the possible number ways\[ = 1\]
Now when
\[P + Q = R + S = 1\]
Where LM, NO= 01, 10
So the possible number ways\[ = 2 \times 2 = 4\]
Now when
\[P + Q = R + S = 2\]
Where LM, NO= 02, 20, 11
So the possible number ways\[ = 3 \times 3 = 9\]
Now when
\[P + Q = R + S = 3\]
Where LM, NO= 03, 30, 21, 12
So the possible number ways\[ = 4 \times 4 = 16\]
Now when
\[P + Q = R + S = 4\]
Where LM, NO= 04, 40, 22, 13, 31
So the possible number ways\[ = 5 \times 5 = 25\]
Now similarly when \[P + Q = R + S = 5\]
So the possible number ways\[ = 6 \times 6 = 36\]
When \[P + Q = R + S = 6\]
So the possible number ways\[ = 7 \times 7 = 49\]
When \[P + Q = R + S = 7\]
So the possible number ways\[ = 8 \times 8 = 64\]
When \[P + Q = R + S = 8\]
So the possible number ways\[ = 9 \times 9 = 81\]
When \[P + Q = R + S = 9\]
So the possible number ways\[ = 10 \times 10 = 100\]
Now when \[P + Q = R + S = 10\]
Where LM, NO= 55, 46, 64, 28, 73, 37, 82, 19, 91
So the possible number ways\[ = 9 \times 9 = 81\]
When \[P + Q = R + S = 11\]
Where LM, NO= 56, 65, 38, 74, 47, 83, 29, 92
So the possible number ways\[ = 8 \times 8 = 64\]
Similarly when \[P + Q = R + S = 12\]
So the possible number ways\[ = 7 \times 7 = 49\]
Now when \[P + Q = R + S = 13\]
So the possible number ways\[ = 6 \times 6 = 36\]
Also when \[P + Q = R + S = 14\]
So the possible number ways\[ = 5 \times 5 = 25\]
Also when \[P + Q = R + S = 15\]
So the possible number ways\[ = 4 \times 4 = 16\]
Also when \[P + Q = R + S = 16\]
So the possible number ways\[ = 3 \times 3 = 9\]
When \[P + Q = R + S = 17\]
So the possible number ways\[ = 2 \times 2 = 4\]
When \[P + Q = R + S = 18\]
So the possible number ways\[ = 1 \times 1 = 1\]
Hence the possible number of suck lucky plate \[ = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1\]
Therefore the total number of suck lucky plate \[ = 670\]

Note: On a number plate each digit can be from 0 to 9, so the sum of the two digits on a number plate can be maximum 18 and minimum 0 and sum of all the four digits will be 36.