Answer

Verified

383.7k+ 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.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE