Question

How do you calculate ten combinations of 10 numbers?

Answer 1

Hint: This sum is formula based for easy solving. While using the formula for finding the 10 combinations, remember that the formula only provides the answer for one type of combination at a time, hence you will have to find the total groups the ten numbers can be combined and then add them.

Complete step-by-step answer:
We can do this sum by doing the combinations of 10 numbers manually or by using the combination formula.
 Using the combination formula is the easy way here. We use the combination formula here since the order is not mentioned. Combination is used to determine the possible arrangements of the items where the order does not matter.
The formula for combination is $ ^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r!} \right)}} $
Where $ ^n{C_r} $ = number of combinations
n = total number of objects
r = number of objects chosen from the set.
Since we have been told to find a combination of 10 numbers, n = 10.
our r will vary from 1 to 10 as this are the possible combinations of 10 numbers and then we will have to add all these.
Hence the total combinations = \[\sum\limits_{r = 1}^{10} {^{10}{C_r}} \]
 $ { = ^{10}}{C_1}{ + ^{10}}{C_2}{ + ^{10}}{C_3}{ + ^{10}}{C_4}{ + ^{10}}{C_5}{ + ^{10}}{C_6}{ + ^{10}}{C_7}{ + ^{10}}{C_8}{ + ^{10}}{C_9}{ + ^{10}}{C_{10}} $
Using the formula for combination we get
Total combination
 $ = \dfrac{{10!}}{{1!\left( {10 - 1!} \right)}} + \dfrac{{10!}}{{2!\left( {10 - 2!} \right)}} + \dfrac{{10!}}{{3!\left( {10 - 3!} \right)}} + \dfrac{{10!}}{{4!\left( {10 - 4!} \right)}} + \dfrac{{10!}}{{5!\left( {10 - 5!} \right)}} + \dfrac{{10!}}{{6!\left( {10 - 6!} \right)}} + \dfrac{{10!}}{{7!\left( {10 - 7!} \right)}} + \dfrac{{10!}}{{8!\left( {10 - 8!} \right)}} + \dfrac{{10!}}{{9!\left( {10 - 9!} \right)}} + \dfrac{{10!}}{{10!\left( {10 - 10!} \right)}} $ solving the above equation we get,
Total combination = 10+45+120+210+252+210+120+45+10+1
=1023
Hence there are total 1023 combinations possible using 10 numbers.
So, the correct answer is “ 1023”.

Note: The formula used above can be only used when the selection is asked without repetition. Combination is often confused with permutation. The only difference in both of these is that the order of the selection matters the most in permutation whereas it does not matter in combination.