Question

The number of six digit numbers in which digits are in ascending order must be:
A. 48
B. 84
C. 120
D. 126

Answer 1

Hint: We know that ascending order means the numbers must increase while going from left to right and we have been asked to arrange the digits in a six digit number in ascending order. Some of these numbers could be: 456789, 345789, 245789 etc.

Complete step-by-step answer:
The six places in the six digit number can be filled by all the nine digits except zero that is the numbers from 1-9. Hence we will be selecting six digits from the nine digits from 1-9 by using the formula for combination which is given by:
$
  ^n{C_r} = \dfrac{{\left| \!{\underline {\,
  n \,}} \right. }}{{\left| \!{\underline {\,
  {n - r} \,}} \right. .\left| \!{\underline {\,
  r \,}} \right. }} \;
$
Similarly we can find the number of six digit numbers in which digits are in ascending order where we know that the value of n is 9 and value of r is 6.
$^9{C_6} = \dfrac{{\left| \!{\underline {\,
  9 \,}} \right. }}{{\left| \!{\underline {\,
  {9 - 6} \,}} \right. .\left| \!{\underline {\,
  6 \,}} \right. }} = \dfrac{{9 \times 8 \times 7 \times \left| \!{\underline {\,
  6 \,}} \right. }}{{\left| \!{\underline {\,
  3 \,}} \right. .\left| \!{\underline {\,
  6 \,}} \right. }}$
Cancelling the factorial 6 from both the numerator and denominator in the equation above we get,
$^9{C_6} = \dfrac{{9 \times 8 \times 7 \times \left| \!{\underline {\,
  6 \,}} \right. }}{{\left| \!{\underline {\,
  3 \,}} \right. .\left| \!{\underline {\,
  6 \,}} \right. }} = \dfrac{{504}}{6} = 84$
Since in this question we have been restricted by stating that the numbers in the six digit number must be arranged in an increasing order therefore we use counting combination and not permutation.
So, the correct answer is “Option B”.

Note: We observe that once we choose any six digits from the nine digits we are only left with one way of arranging it since they must be increasing in order while placing. Therefore we choose six digits at a time from the nine digits and permutation does not come into play because we can not arrange it in more than one way.