Calculus Or Combinatorics ?

Calculus Level 4

$\large f(x) =g_{1}(x)\cdot g_{2}(x)\cdots g_{19}(x) \cdot g_{20}(x)$

Consider a function $f(x)$ , which can be expressed as product of 20 different smooth functions. Find the number of all different terms in the 10th derivative of $f(x)$ .

\dbinom{29}{10}

\dbinom{28}{10}

None of the others

\dbinom{30}{10}

\dbinom{31}{10}

1 solution

Bob Bob
Oct 7, 2017

It is clear from the product rule that an arbitrary term of the tenth derivative is of the form $\displaystyle k {g_1}^{(k_1)}(x) {g_2}^{(k_2)}(x) \dots {g_{20}}^{(k_{20})}(x)$ for some positive integers $\displaystyle k, \ k_1, \ \dots , \ k_{20}$ . But note that these indices $\displaystyle k_i$ together sum to 10, since this term has been arrived at by taking ten derivatives. So this problem is equivalent to finding the number of multisets of size ten with elements taken from a set of size twenty, which is $\displaystyle {{20 + 10 - 1} \choose 10} = {29 \choose 10}$ .

Calculus Or Combinatorics ?

1 solution

0 pending reports