Question

What is the value of $^8{c_4}{ + ^8}{c_3}$ ?
A. $^8{c_5}$
B. 63
C. 35
D. $^9{c_4}$

Answer 1

Combination helps to determine the number of possible arrangements in a collection of items where the order of selection is not important. It can be calculated by using the formula,
$^n{c_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} +$
Where, n-total number of items
r- number of selecting objects from the given set

Complete step by step answer:
Factorial (n!) is defined as the product of all positive integers from 1 up to n. Combination can be calculated by using the formula,
$^n{c_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} +$
Where,
n-total number of items
r- number of selecting objects from the given set
Thus, from combination formula, $^8{c_4}$ can be calculated as,
$^8{c_4} = \dfrac{{8!}}{{4!\left( {8 - 4} \right)!}}$
${ \Rightarrow ^8}{c_4} = \dfrac{{8!}}{{4!4!}}$
${ \Rightarrow ^8}{c_4} = \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{1 \times 2 \times 3 \times 4 \times 1 \times 2 \times 3 \times 4}}$
${ \Rightarrow ^8}{c_4} = 70$
Similarly, $^8{c_3}$ can be calculated as,
$^8{c_3} = \dfrac{{8!}}{{3!\left( {8 - 3} \right)!}}$
${ \Rightarrow ^8}{c_3} = \dfrac{{8!}}{{3!5!}}$
${ \Rightarrow ^8}{c_3} = \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{1 \times 2 \times 3 \times 1 \times 2 \times 3 \times 4 \times 5}}$
${ \Rightarrow ^8}{c_4} = 56$
Thus, $^8{c_4}{ + ^8}{c_3} = 70 + 56$
${ \Rightarrow ^8}{c_4}{ + ^8}{c_3} = 126$
Among four options, $^9{c_4}$ has value 126 which is similar to $^8{c_4}{ + ^8}{c_3}$
${ \Rightarrow ^8}{c_4}{ + ^8}{c_3}{ = ^9}{c_4}$

Thus, option D $^9{c_4}$4 is correct.

Note: This question can also be approached in an easier way.
Generally, $^n{c_r}{ + ^n}{c_{r - 1}}{ = ^{n + 1}}{c_r}$
Here, n=8,r=4
Therefore, the answer for $^8{c_4}{ + ^8}{c_3}{ = ^{8 + 1}}{c_3}$
${ \Rightarrow ^8}{c_4}{ + ^8}{c_3}{ = ^9}{c_4}$
Combinations and permutations are the way of representing a group of objects by selecting them in a set and thus., forming subsets. Combination helps to determine the number of possible arrangements in a collection of items where the order of selection is not important whereas permutation helps to determine the number of possible arrangements in a collection of items in an ordered manner. Zero factorial, 0! is equal to one. Binomial is a polynomial with two terms. Binomial theorem explains the algebraic expression of power of a binomial. Binomial theorem helps to prove results in calculus, differentials and combinatorics. Binomial theorem is also helpful to find the probability in an organised way.