14 pegs removed in toto...

111 brute force computer simulations run before following solution found...

$logged (array)
0 Move 1: Peg #4 removed to begin play (creates initial hole)...
1 Move 2: Peg #11 moved to hole #4 and jumped peg #7 removed.
2 Move 3: Peg #2 moved to hole #7 and jumped peg #4 removed.
3 Move 4: Peg #13 moved to hole #4 and jumped peg #8 removed.
4 Move 5: Peg #10 moved to hole #8 and jumped peg #9 removed.
5 Move 6: Peg #15 moved to hole #13 and jumped peg #14 removed.
6 Move 7: Peg #12 moved to hole #14 and jumped peg #13 removed.
7 Move 8: Peg #4 moved to hole #13 and jumped peg #8 removed.
8 Move 9: Peg #13 moved to hole #15 and jumped peg #14 removed.
9 Move 10: Peg #3 moved to hole #10 and jumped peg #6 removed.
10 Move 11: Peg #15 moved to hole #6 and jumped peg #10 removed.
11 Move 12: Peg #6 moved to hole #4 and jumped peg #5 removed.
12 Move 13: Peg #7 moved to hole #2 and jumped peg #4 removed.
13 Move 14: Peg #1 moved to hole #4 and jumped peg #2 removed.

Only peg in hole #4 remained at run's end...

[Note: All moves, including initial one, were randomly selected.]