Combinatorics, EXT1 A1 EQ-Bank 8 Show `\ ^nC_k = \ ^nC_(n-k)`. (1 mark) Show Answers Only `text(See Worked Solutions)` Show Worked Solution `\ ^nC_k = (n!)/((n-k)!k!)` `\ ^nC_(n-k)` `= (n!)/((n-(n-k))!(n-k)!)` `= (n!)/(k!(n-k)!)` `= \ ^nC_k`
Combinatorics, EXT1 A1 EQ-Bank 7 Show `\ ^nC_k = \ ^(n-1)C_(k-1) + \ ^(n-1)C_k`. (2 marks) --- 6 WORK AREA LINES (style=lined) --- Show Answers Only `text(See Worked Solutions)` Show Worked Solution `text(LHS) = (n!)/((n-k)!k!)` `text(RHS)` `= ((n-1)!)/((n-1-(k-1))!(k-1)!) + ((n-1)!)/((n-1-k)!k!)` `= ((n-1)!k)/((n-k)!(k-1)!k) + ((n-1)!(n-k))/((n-k-1)!(n-k)k!)` `= ((n-1)!k)/((n-k)!k!) + ((n-1)!(n-k))/((n-k)!k!)` `= ((n-1)!(k + n-k))/((n-k)!k!)` `= (n!)/((n-k)!k!)` `=\ text(LHS)`