The clue is the two 14's in the ratios. Pascal's triangle, which is related to binomial expansions, which in turn include nCr values, tells us that n must be even and if two identical terms in the binomial coefficients are separated by two as r+2 and r+4 are, the values r+2 and r+4 must be complementary, they must add up to n. So we can relate r and n, r+2+r+4=n, making n=2(r+3). The difference between the binomial coefficients for r and r+2 is (n-r)(n-r-1)/(r+1)(r+2) and the ratio between these is 14:3. First substitute for n, giving us (r+6)(r+5) as the numerator, then apply the ratio, so 14(r+1)(r+2)=3(r+6)(r+5). This reduces to the quadratic 11r^2+9r-62=0, which factorises to (11r+31)(r-2)=0. The only sensible value is r=2, so n=10. The three nCr's are 45, 210 and 210, in the ratios 3:14:14.
NOTE: The binomial coefficients are calculated from (n-1)(n-2)...(n-r+1)/1*2*3...r. This is also how nCr's are calculated. An inspection of the 10th row of Pascal's triangle confirms the answer.