The Gov.uk Holiday Entitlement Calculator makes it 9.8 days.
Your entitlement for the Holiday Year is: 5.6 weeks x days worked per week. Therefore for the year, the it is 22.4 days.
Your Holiday Year started on 4th June 2012 unless your contract says otherwise.
If you work on a Monday then the following Bank Holidays you may or may not have worked.
04 June Monday Spring bank holiday
05 June Tuesday Queen?s Diamond Jubilee
27 August Monday Summer bank holiday
Do you have any notes about what days you actually worked when you first started your job, as there are two Bank Holidays in your first week of work.
Unless your contract says otherwise, your holiday entitlement (22.4 days) includes Bank Holidays that fall on a working day. So if they were your working days and if you did not go into work but have been paid for those days, then they come out of your holiday entitlement. Does your 10 days off include any Bank Holidays that were on a working day?
If your first day of work was Monday 4th June and your last day of work is Friday 9th November, and you work 4 days per week, then I make that 92 working days. If you are working different hours on different days, then holiday may be calculated in hours rather than days.
If you did 40 hours a week, then that's 920 hours for the 23 full weeks you have worked. Divide by 8.29 (or multiply by 12.07%), gives you 110.98 hours. You would then deduct the hours from that, as per what your working hours would have been on the days you took as holiday.
Has your employer told you how they have calculated your holiday entitlement, thus why they think you owe them 4 days?