Number of ways a black square can be chosen = 32.
Number of ways a white square can be chosen which is not in same
row or column as the black square = 24.
So the total number of ways = 32 × 24 = 768.
Last two digit should be 12, 16, 24, 32, 36, 52, 56, 64 – 8 ways
Remaining three places 4 × 3 × 2 = 24
Total 8 × 24 = 192 ways.
he phone number is of the form
635 _ _ _ _ or 674 _ _ _ _
and the digit 9 is used exactly once.
So the problem is reduced to identifying a 4 digit odd number which uses 9 exactly once.
_ _ _ _
when 9 is at the thousand's or hundreds or ten's place, the other three digits can be filled in 9, 9 and 4 (for
units place) ways respectively.
No. of numbers
and when 9 is at unit's place, the number is definitely odd. The total numbers possible is this case is
total number of number that can be formed
= 4 × 9 × 9 × 3 + 9 × 9 × 9 = 1701
but 1701 numbers are possible for both telephone numbers starting with 635 and 674.
∴Total number of tries
= 2 × 1701 = 3402 .
Since we need 6 digit palindrome we only need to identify 3 digits
– – – / – – –
out of these three, digit at unit’s place can be selected in 4 ways (since it should be an even number and 0 cannot come) and other two digits can be selected in 10 ways so total number of ways
10 × 10 × 4 = 400.
Consider when 4 comes at unit's place.
total number of 4 digit numbers that can be formed
4
2 2 1
= 2 × 2 = 4
Thus, the digit 4 will come at the unit's place 4 times.
Same can be said of digits 1 and 2.
∴ sum of unit's digit column (when all possible numbers of 4 digit using 0, 1, 2 are written one below the
other)
= 4 × 4 + 4 × 2 + 4 × 1 = 28
same will be for hundred's and ten's column.
In case of 1, 2 or 4 coming in thousand's place, total number of numbers that can be formed
⇒
4
3 2 1
= 3 × 2 = 6
in thousand's column, 4 will come 6 times and so will and 2.
We can now write the sum as.
Thousand’s
4 5
6×4
6×2
6×1
Hundred’s
3 1
4×4
4×2
4×1
Ten’s
3 0
4×4
4×2
4×1
One’s
2 8
4×4
4×2
4×1
3 3 2 (note: zeros may
come in any column
any number of
times and still will
not affect the sum)
Thus the answer is 45108.