Mathematics - Permutation & Combination Question with Solution | TestHub

$S_i=\left\{\left(i,j,k\right):i,j,k\in \left\{1,2,\mathrm{...},10\right\}\right\}$

$n_1=10\times 10\times 10=1000$

i.e., number of elements in $S_1=1000$

$S_2=\left\{\left(i,j\right):1\le i<j+2\le 10,i,j\in \left\{1,2,\mathrm{...},10\right\}\right\}$

$n_2=2+3+4+5+6+7+8+9$

$=44$

$S_3=\left\{\left(i,j,k,l\right):1\le i<j<k<l,i,j,k,l\in \left\{1,2,\mathrm{....10}\right\}\right\}$

$n_3{=}^{10}{C}_4$

$=\frac{10\times 3\times 8\times 7\times 6!}{4\times 3\times 2\times 1\times 6!}=210$

i.e. number of elements in $S_3=210$

$S_4=\left\{\left(i,j,k,l\right):i,j,k \text{and} l \text{are distinct elements in} \left\{1,2,\mathrm{.......},10\right\}\right\}$

$n_4=10\times 9\times 8\times 7$ or ${}^{10}{C}_4\times 4!$

$=5040$

$\frac{n_4}{12}=\frac{5040}{12}=420$