Question
In the figure, two 4-digit numbers are to be formed by filling the places with digits. The number of different ways in which the places can be filled by digits so that the sum of the numbers formed is also a 4-digit number and in no place the addition is with carrying, is
In the figure, two 4-digit numbers are to be formed by filling the places with digits. The number of different ways in which the places can be filled by digits so that the sum of the numbers formed is also a 4-digit number and in no place the addition is with carrying, is
A.
$${55^4}$$
B.
$$220$$
C.
$${45^4}$$
D.
None of these
Answer :
None of these
Solution :
If 0 is placed in the units place of the upper number then the units place of the lower number can be filled in 10 ways (filling by any one of 0, 1, 2, . . . . . , 9).
If 1 is placed in the units place of the upper number then the units place of the lower number can be filled in 9 ways (filling by any one of 0, 1, 2, . . . . . , 8), e.t.c.
∴ the units column can be filled in $$10 + 9 + 8 + . . . . . + 1,$$ i.e., 55 ways.
Similarly for the second and the third columns. The number of ways for the fourth column $$= 8 + 7 + . . . . . + 1 = 36$$
∴ the required number of ways $$ = 55 \times 55 \times 55 \times 36.$$
If 0 is placed in the units place of the upper number then the units place of the lower number can be filled in 10 ways (filling by any one of 0, 1, 2, . . . . . , 9).
If 1 is placed in the units place of the upper number then the units place of the lower number can be filled in 9 ways (filling by any one of 0, 1, 2, . . . . . , 8), e.t.c.
∴ the units column can be filled in $$10 + 9 + 8 + . . . . . + 1,$$ i.e., 55 ways.
Similarly for the second and the third columns. The number of ways for the fourth column $$= 8 + 7 + . . . . . + 1 = 36$$
∴ the required number of ways $$ = 55 \times 55 \times 55 \times 36.$$