
How many $6$ digit numbers can be formed using the digits $1,2,3$ only, whose sum of digits is $10?$
Answer
433.5k+ views
Hint:As we can see in the above question we have to use the combination because we have to choose the numbers to make six digits in such a way that the sum has to be $10$. We know that the selection of some or all objects from a given set of different objects is called combination. The formula of combination is written as $^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$.
Complete step by step answer:
We have been given numbers: $1,2,3$. We have to form six digits from these so that the sum is $10$. Let us make the first selection i.e. $2,2,2,2,1,1$. Now we make the second selection: $(3,3,1,1,1,1)$. We can see that there can be no more selection of numbers from these given numbers.
In the first order we have $2,2,2,2,1,1$. So we have $n = 6$, We have $4$ repeated four times, so $r = 4$. Now we can put this in the formula $\dfrac{{6!}}{{4!(6 - 4)!}}$.
On solving we have,
$\dfrac{{6!}}{{4!2!}} = \dfrac{{6 \times 5 \times 4!}}{{4! \times 2 \times 1}}$.
Further simplifying we have the value: $\dfrac{{6 \times 5}}{2} = 15$.
Similarly we can solve the second selection: $(3,3,1,1,1,1)$.
So we have $n = 6$, We have $1$ repeated four times, so $r = 4$. Now we can put this in the formula $\dfrac{{6!}}{{4!(6 - 4)!}}$.
On solving we have,
$\dfrac{{6!}}{{4!2!}} = \dfrac{{6 \times 5 \times 4!}}{{4! \times 2 \times 1}}$
Further simplifying we have the value: $\dfrac{{6 \times 5}}{2} = 15$.
So we have a total two ways, we add them to get the total number i.e. $15 + 15 = 30$.
Hence the required answer is $30$.
Note:We should be careful about what we have to use in the solution i.e. permutation or combination. When we arrange objects in a definite order then we call it Permutation. The formula of Permutation is written as $^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$. In permutation we know that $0 < r \leqslant n$, it should be cross checked to avoid calculation errors.
Complete step by step answer:
We have been given numbers: $1,2,3$. We have to form six digits from these so that the sum is $10$. Let us make the first selection i.e. $2,2,2,2,1,1$. Now we make the second selection: $(3,3,1,1,1,1)$. We can see that there can be no more selection of numbers from these given numbers.
In the first order we have $2,2,2,2,1,1$. So we have $n = 6$, We have $4$ repeated four times, so $r = 4$. Now we can put this in the formula $\dfrac{{6!}}{{4!(6 - 4)!}}$.
On solving we have,
$\dfrac{{6!}}{{4!2!}} = \dfrac{{6 \times 5 \times 4!}}{{4! \times 2 \times 1}}$.
Further simplifying we have the value: $\dfrac{{6 \times 5}}{2} = 15$.
Similarly we can solve the second selection: $(3,3,1,1,1,1)$.
So we have $n = 6$, We have $1$ repeated four times, so $r = 4$. Now we can put this in the formula $\dfrac{{6!}}{{4!(6 - 4)!}}$.
On solving we have,
$\dfrac{{6!}}{{4!2!}} = \dfrac{{6 \times 5 \times 4!}}{{4! \times 2 \times 1}}$
Further simplifying we have the value: $\dfrac{{6 \times 5}}{2} = 15$.
So we have a total two ways, we add them to get the total number i.e. $15 + 15 = 30$.
Hence the required answer is $30$.
Note:We should be careful about what we have to use in the solution i.e. permutation or combination. When we arrange objects in a definite order then we call it Permutation. The formula of Permutation is written as $^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$. In permutation we know that $0 < r \leqslant n$, it should be cross checked to avoid calculation errors.
Recently Updated Pages
Master Class 9 General Knowledge: Engaging Questions & Answers for Success

Earth rotates from West to east ATrue BFalse class 6 social science CBSE

The easternmost longitude of India is A 97circ 25E class 6 social science CBSE

Write the given sentence in the passive voice Ann cant class 6 CBSE

Convert 1 foot into meters A030 meter B03048 meter-class-6-maths-CBSE

What is the LCM of 30 and 40 class 6 maths CBSE

Trending doubts
Which one is a true fish A Jellyfish B Starfish C Dogfish class 11 biology CBSE

What is the difference between superposition and e class 11 physics CBSE

State and prove Bernoullis theorem class 11 physics CBSE

1 ton equals to A 100 kg B 1000 kg C 10 kg D 10000 class 11 physics CBSE

State the laws of reflection of light

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
